Differential Geometry III Lecture Notes - A. Kovalev

Part III: Differential geometry (Michaelmas 2004)

Alexei Kovalev ([email protected])

The line and surface integrals studied in vector calculus require a concept of a curve and asurface in Euclidean 3-space. These latter objects are introduced via a parameterization:a smooth map of, respectively, an interval on the real line R or a domain in the plane R2

into R3 . In fact, one often requires a so-called regular parameterization. For a curve r(t),this means non-vanishing of the ‘velocity vector’ at any point, r(t) 6= 0. On a surface apoint depends on two parameters r = r(u, v) and regular parameterization means that thetwo vectors of partial derivatives in these parameters are linearly independent at everypoint ru(u, v)× rv(u, v) 6= 0.

Differential Geometry develops a more general concept of a smooth n-dimensionaldifferentiable manifold.1 and a systematic way to do differential and integral calculus (andmore) on manifolds. Curves and surfaces are examples of manifolds of dimension d = 1and d = 2 respectively. However, in general a manifold need not be given (or considered)as lying in some ambient Euclidean space.

1.1 Manifolds: definitions and first examples

The basic idea of smooth manifolds, of dimension d say, is to introduce a class of spaceswhich are ‘locally modelled’ (in some precise sense) on a d-dimensional Euclidean spaceRd.

A good way to start is to have a notion of open subsets (sometimes one says ‘to havea topology’) on a given set of points (but see Remark on page 2).

Definition. 1. A topological space is a set, M say, with a specified class of opensubsets, or neighbourhoods, such that

(i) ∅ and M are open;

(ii) the intersection of any two open sets is open;

(iii) the union of any number of open sets is open.

2. A topological space M is called Hausdorff if any two points of M possess non-intersecting neighbourhoods.

3. A topological space M is called second countable if one can find a countablecollection B of open subsets of M so that any open U ⊂M can be written as a unionof sets from B.

The last two parts of the above definition will be needed to avoid some pathologicalexamples (see below).

1More precisely, it is often useful to also consider appropriate ‘structures’ on manifolds (e.g. Riemannianmetrics), as we shall see in due course.

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The knowledge of open subsets enables one to speak of continuous maps: a map betweentopological spaces is continuous if the inverse image of any open set is open. Exercise: checkthat for maps between open subsets (in the usual sense) of Euclidean spaces this definitionis equivalent to other definitions of a continuous map.

A homeomorphism is a bijective continuous map with continuous inverse. More ex-plicitly, to say that ‘a bijective mapping ϕ of U onto V is a homeomorphism’ means that‘D ⊂ U is open if and only if ϕ(D) ⊂ V is open’.

Let M be a topological space. A homeomorphism ϕ : U → V of an open set U ⊆ Monto an open set V ⊆ Rd will be called a local coordinate chart (or just ‘a chart’) andU is then a coordinate neighbourhood (or ‘a coordinate patch’) in M .

Definition. A C∞ differentiable structure, or smooth structure, on M is a collectionof coordinate charts ϕα : Uα → Vα ⊆ Rd (same d for all α’s) such that

(i) M = ∪α∈AUα;

(ii) any two charts are ‘compatible’: for every α, β the change of local coordinates ϕβϕ−1α

is a smooth (C∞) map on its domain of definition, i.e. on ϕα(Uβ ∩ Uα) ⊆ Rd.

(iii) the collection of charts ϕα is maximal with respect to the property (ii): if a chart ϕof M is compatible with all ϕα then ϕ is included in the collection.

A bijective smooth map with a smooth inverse is called a diffeomorphism. Noticethat clause (ii) in the above definition implies that any change of local coordinates is adiffeomorphism between open sets ϕα(Uβ ∩ Uα) and ϕβ(Uβ ∩ Uα) of Rd.

In practice, one only needs to worry about the first two conditions in the above defini-tion. Given a collection of compatible charts covering M , i.e. satisfying (i) and (ii), thereis a unique way to extend it to a maximal collection to satisfy (iii). I leave this last claimwithout proof but refer to Warner, p. 6.

Definition . A topological space equipped with a C∞ differential structure is called asmooth manifold. Then d is called the dimension of M , d = dimM .

Sometimes in the practical examples one starts with a differential structure on a set ofpoints M (with charts being just bijective maps) and then defines the open sets in M tobe precisely those making the charts into homeomorphisms. More explicitly, one then saysthat D ⊂ M is open if and only if for every chart ϕ : U → V ⊆ Rd, ϕ(D ∩ U) is openin Rd. We shall refer to this as the topology induced by a C∞ structure.

Remarks . 1. Some variations of the definition of the differentiable structure are possible.Much of the material in these lectures could be adapted to Ck rather than C∞ differentiablemanifolds, for any integer k > 0. 2

On the other hand, replacing Rd with the complex coordinate space Cn and smoothmaps with holomorphic (complex analytic) maps between domains in C, leads of an im-portant special class of complex manifolds—but that is another story.

2if k = 0 then the definition of differentiable structure has no content

differential geometry 3

By a manifold I will always mean in these lectures a smooth (real) manifold, unlessexplicitly stated otherwise.

2. Here is an example of what can happen if one omits a Hausdorff property. Considerthe following ‘line with a double point’

M = (−∞, 0) ∪ 0′, 0′′ ∪ (0,∞),

with two charts being the obvious ‘identity’ maps (the induced topology is assumed)ϕ1 : U1 = (−∞, 0) ∪ 0′ ∪ (0,∞) → R, ϕ2 : U2 = (−∞, 0) ∪ 0′′ ∪ (0,∞) → R, soϕ1(0

′) = ϕ1(0′′) = 0. It is not difficult to check that M satisfies all the conditions of a

smooth manifold, except for the Hausdorff property (0′ and 0′′ cannot be separated).

Omitting the 2nd countable property would allow, e.g. an uncountable collection (dis-joint union) of lines

t0<α<1 Rα,

each line Rα equipped with the usual topology and charts being the identity maps Rα → R.3

Examples. The Euclidean Rd is made into a manifold using the identity chart. Thecomplex coordinate space Cn becomes a 2n-dimensional manifold via the chart Cn → R2n

replacing every complex coordinate zj by a pair of real coordinates Re zj, Im zj.The sphere Sn = x ∈ Rn+1 :

∑ni=0 x

2i = 1 is made into a smooth manifold of

dimension n, by means of the two stereographic projections onto Rn ∼= x ∈ Rn+1 :x0 = 0, from the North and the South poles (±1, 0, . . . , 0). The corresponding change ofcoordinates is given by (x1, . . . , xn) 7→ (x1/|x|2, . . . , xn/|x|2).

The real projective space RP n is the set of all lines in Rn+1 passing through 0. Elementsof RP n are denoted by x0 : x1 : . . . : xn, where not all xi are zero. Charts can be given byϕi(x0 : x1 : . . . : xn) = (x0/xi, . . . i . . . , xn/xi) ∈ Rn with changes of coordinates given by

ϕj ϕ−1i : (y1, . . . , yn) 7→ y1 : . . . : (1 in ith place) : . . . : yn ∈ RP n 7→(

y1

yj

, . . . ,yi−1

yj

,1

yj

,yi

yj

, . . . ,yj−1

yj

,yj+1

yj

. . .yn

yj

),

smooth functions on their domains of definition (i.e. for yj 6= 0). Thus RP n is a smoothn-dimensional manifold.

Definition. Let M,N be smooth manifolds. A continuous map f : M → N is calledsmooth (C∞) if for each p ∈ M , for some (hence for every) charts ϕ and ψ, of M andN respectively, with p in the domain of ϕ and f(p) in the domain of ψ, the compositionψ f ϕ−1 (which is a map between open sets in Rn, Rk, where n = dimM , k = dimN)is smooth on its domain of definition.

Exercise: write out the domain of definition for ψ f ϕ−1.

Two manifolds M and N are called diffeomorphic is there exists a smooth bijectivemap M → N having smooth inverse. Informally, diffeomorphic manifolds can be thoughtof as ‘the same’.

3For a more interesting example on a ‘non 2nd countable manifold’ see Example Sheet Q1.10.

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1.2 Matrix Lie groups

Consider the general linear group GL(n,R) consisting of all the n × n real matrices Asatisfying detA 6= 0. The function A 7→ detA is continuous and GL(n,R) is the inverseimage of the open set R \ 0, so it is an open subset in the n2-dimensional linear space ofall the n × n real matrices. Thus GL(n,R) is a manifold of dimension n2. Note that theresult of multiplication or taking the inverse depends smoothly on the matrix entries.

Similarly, GL(n,C) is a manifold of dimension 2n2 (over R).

Definition. A group G is called a Lie group if it is a smooth manifold and the map(σ, τ) ∈ G×G→ στ−1 ∈ G is smooth.

Let A be an n× n complex matrix. The norm given by |A| = nmaxij |aij| has a usefulproperty that |AB| ≤ |A||B| for any A,B. The exponential map on the matrices is definedby

exp(A) = I + A+ A2/2! + . . .+ An/n! + . . . .

The series converge absolutely and uniformly on any set |A| ≤ µ, by the WeierstrassM-test. It follows that e.g. exp(At) = (exp(A))t and exp(C−1AC) = C−1 exp(A)C,for any invertible matrix C. Furthermore, the term-by-term differentiated series also con-verge uniformly and so exp(A) is C∞-smooth in A. (This means smooth as a function of2n2 real variables, the entries of A.)

The logarithmic series

log(I + A) = A− A2/2 + . . .+ (−1)n+1An/n+ . . .

converge absolutely for |A| < 1 and uniformly on any closed subset |A| ≤ ε, for ε < 1,and log(A) is smooth in A.

One has

exp(log(A)) = A, when |A− I| < 1. (1.1)

This is true in the formal sense of composing the two series in the left-hand side. Theformal computations are valid in this case as the double-indexed series in the left-handside is absolutely convergent.

For the other composition, one has

log(exp(A)) = A when |A| < log 2. (1.2)

again by considering a composition of power series with a similar reasoning.

Remark . Handling the power series of complex matrices in (1.1) and (1.2) is quite similarto handling 1 × 1 matrices, i.e. complex numbers. Warning: not all the usual propertiescarry over wholesale, as the multiplication of matrices is not commutative. E.g., in general,exp(A) exp(B) 6= exp(A + B). However, the identity exp(A) exp(−A) = I does hold (andthis is used in the proof of Proposition 1.3 below).


Proposition 1.3. The orthogonal group O(n) = A ∈ GL(n,R) : AAt = I has a smoothstructure making it into a manifold of dimension n(n− 1)/2.

The charts take values in the n(n−1)2

-dimensional linear space of skew-symmetric n× nreal matrices. E.g.

ϕ : A ∈ A ∈ O(n) : |A− I| is small 7→ B = log(A) ∈ B : Bt = −B

is a chart in a neighbourhood of I ∈ O(n). The desired smooth structure is generated bya family of charts of the form ϕC(A) = log(C−1A), where C ∈ O(n).

The method of proof of Proposition 1.3 is not specific to the orthogonal matrices andworks for many other subgroups of GL(n,R) or GL(n,C) (Example Sheet 1, Question 4).

1.3 Tangent space to a manifold

If x(t) is a smooth regular curve in Rn then the velocity vector x(0) is a tangent vector tothis curve at t = 0. In a change of coordinates x′i = x′i(x) the coordinates of this vector aretransformed according to the familiar chain rule, applied to x′(x(t)). One consequence isthat the statement “two curves pass through the same point with the same tangent vector”is independent of the choice of coordinates, that is to say a tangent vector (understood asvelocity vector) is a geometric object.

The above observation is local (depends only on what happens in a neighbourhoodof a point of interest), therefore the definition may be extended to an arbitrary smoothmanifold.

Definition. A tangent vector to a manifold M at a point p ∈ M is a map a assigning toeach chart (U,ϕ) with p ∈ U an element in the coordinate space (a1, a2, . . . , an) ∈ Rn insuch a way that if (U ′, ϕ′) is another chart then

a′i =

(∂x′i∂xj

)p

aj, (1.4)

where xi, x′i are the local coordinates on U,U ′ respectively. 4 All the tangent vectors at a

given point p form the tangent space denoted TpM .

Remarks. It is easy to check that TpM is naturally a vector space.The transformation law (1.4) is the defining property of a tangent vector.

Notation. A choice local coordinates xi on a neighbourhood U ⊆ M defines a linearisomorphism TpM → Rn. A basis of TpM corresponding to the standard basis of Rn viathis isomorphism is a usually denoted by ( ∂

∂xi)p. The expression of a tangent vector in

local coordinates a(U,ϕ) = (a1, . . . , an) then becomes ai(∂

∂xi)p.

4Here and below the convention is used that if the same letter appears as an upper and lower indexthen the summation is performed over the range of this index. E.g. the summation in j = 1, . . . , n in thisinstance.

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As can be seen from (1.4), the standard basis vectors of TpM given by the local coor-dinates xi and x′i are related by(

∂

∂xi

)p

=

(∂x′j∂xi

)p

(∂

∂x′j

)p

. (1.4′)

This is what one would expect in view of the chain rule from the calculus. The formula (1.4′)tells us that every tangent vector ai(

∂∂xi

)p gives a well-defined first-order ‘derivation’

ai(∂

∂xi

)p : f ∈ C∞(M) → ai∂f

∂xi

(p) ∈ R. (1.5)

In fact the following converse statement is true although I shall not prove it here5. Givenp ∈M , every linear map a : C∞(M) → R satisfying Leibniz rule a(fg) = a(f)g(p) +f(p)a(g) arises from a tangent vector as in (1.5).

Example 1.6. Let r = r(u, v), (u, v) ∈ U ⊆ R2 be a smooth regular-parameterized surfacein R3. Examples of tangent vectors are the partial derivatives ru, rv —these correspondto just ∂

∂u, ∂∂v

in the above notation, as the parameterization by u, v is an instance of acoordinate chart.

Definition. A vector space with a multiplication [·, ·], bilinear in its arguments (thus satis-fying the distributive law), is called a Lie algebra if the multiplication is anti-commutative[a, b] = −[b, a] and satisfies the Jacobi identity [[a, b], c] + [[b, c], a] + [[c, a], b] = 0.

Theorem 1.7 (The Lie algebra of a Lie group). Let G be a Lie group of matrices andsuppose that log defines a coordinate chart near the identity element of G. Identify thetangent space g = TIG at the identity element with a linear subspace of matrices, via thelog chart, and then g is a Lie algebra with [B1, B2] = B1B2 −B2B1.

The space g is called the Lie algebra of G.

Proof. It suffices to show that for every two matrices B1, B2 ∈ g, the [B1, B2] is also anelement of g. As [B1, B2] is clearly anticommutative and the Jacobi identity holds formatrices, g will then be a Lie algebra.

The expression

A(t) = exp(B1t) exp(B2t) exp(−B1t) exp(−B2t)

defines, for |t| < ε with sufficiently small ε, a path A(t) in G such that A(0) = I. Usingfor each factor the local formula exp(Bt) = I +Bt+ 1

2B2t2 + o(t2), as t→ 0, 6 we obtain

A(t) = I + [B1, B2]t2 + o(t2), as t→ 0.

5A proof can be found in Warner 1.14-1.20. Note that his argument requires C∞ manifolds (and doesnot extend to Ck).

6The notation o(tk) means a remainder term of order higher than k, i.e. r(t) such that limt→0 r(t)/tk = 0.


Hence

B(t) = logA(t) = [B1, B2]t2 + o(t2) and exp(B(t) = A(t)

hold for any sufficiently small |t| and so B(t) ∈ g (as B(t) is in the image of the logchart). Hence B(t)/t2 ∈ g for every small t 6= 0 (as g is a vector space). But thenalso limt→0B(t)/t2 = limt→0([B1, B2] + o(1)) = [B1, B2] ∈ g (as g is a closed subset ofmatrices).

Notice that the idea behind the above proof is that the Lie bracket [B1, B2] on a Liealgebra g is an ‘infinitesimal version’ of the commutator g1g2g

−11 g−1

2 in the correspondingLie group G.

Definition. Let M be a smooth manifold. A disjoint union TM = tp∈MTpM is called thetangent bundle of M .

Theorem 1.8 (The ‘manifold of tangent vectors’). The tangent bundle TM has acanonical differentiable structure making it into a smooth 2n-dimensional manifold, wheren = dimM .

The charts identify any tp∈UTpM ⊆ TM , for an coordinate neighbourhood U ⊆ M , withU × Rn. 7 Exercise: check that TM is Hausdorff and second countable (if M is so).

Definition. A (smooth) vector field on a manifold M is a map X : M → TM , such that(i) X(p) ∈ TpM for every p ∈M , and(ii) in every chart, X is expressed as ai(x)

∂∂xi

with coefficients ai(x) smooth functionsof the local coordinates xi.

Theorem 1.9. Suppose that on a smooth manifold M of dimension n there exist n vectorfields X(1), X(2), . . . , X(n), such that X(1)(p), X(2)(p), . . . , X(n)(p) form a basis of TpM atevery point p of M . Then TM is isomorphic to M × Rn.

Here ‘isomorphic’ means that TM and M ×Rn are diffeomorphic as smooth manifoldsand for every p ∈M , the diffeomorphism restricts to an isomorphism between the tangentspace TpM and vector space p ×Rn. (Later we shall make a more systematic definitionincluding this situation as a special case.)

Proof. Define π : ~a ∈ TM → p ∈ M if ~a ∈ TpM ⊂ TM . On the other hand, for any~a ∈ TM , there is a unique way to write ~a = aiX

(i), for some ai ∈ R. Now define

Φ : ~a ∈ TM → (π(~a); a1, . . . , an) ∈M × Rn.

It is clear from the construction and the hypotheses of the theorem that Φ is a bijectionand Φ converts every tangent space into a copy of Rn. It remains to show that Φ and Φ−1

are smooth.

7The topology on TM is induced from the smooth structure.

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Using an arbitrary chart ϕ : U ⊆M → Rn, and the corresponding chart ϕT : π−1(U) ⊆TM → Rn × Rn one can locally express Φ as

(ϕ, idRn) Φ ϕ−1T : (x, (bj)) ∈ ϕ(U)× Rn → (x, (ai)) ∈ ϕ(U)× Rn,

where xi are local coordinates on U , and ~a = bj∂

∂xj. Writing X(i) = X

(i)j (x) ∂

∂xj, we obtain

bj = aiX(i)j (x), which shows that Φ−1 is smooth. The matrix X(x) = (X

(i)j (x)) expresses a

change of basis of TpM , from ( ∂∂xj

)p to X(j)(p), and is smooth in x, so the inverse matrix

C(x) is smooth in x too. Therefore ai = bjCji (x) verifies that Φ is smooth.

Remark . The hypothesis of Theorem 1.9 is rather restrictive. In general, a manifold neednot admit any non-vanishing smooth (or even continuous) vector fields at all (as we shallsee, this is the case for any even-dimensional sphere S2n) and the tangent bundle TM willnot be a product M × Rn.

Definition. The differential of a smooth map F : M → N at a point p ∈ M is a linearmap

(dF )p : TpM → TF (p)N

given in any chart by (dF )p : ( ∂∂xi

)p 7→ (∂yj

∂xi)(p) ( ∂

∂yj)F (p). Here xi are local coordinates

on M , yj on N , and for j = 1, . . . , dimN , yj = Fj(x1, . . . , xn) (n = dimM) give theexpression ψ F ϕ−1 for F in these local coordinates.

It follows, by direct calculations in local coordinates, that the differential is independentof the choice of charts and that the chain rule d(F2 F1)p = (dF2)g(p) (dF1)p holds.

Every (smooth) vector field, say X on M , defines a linear differential operator of firstorder X : C∞(M) → C∞(M), according to (1.5) (with p allowed to vary in M). Supposethat F : M → N is a diffeomorphism. Then for each vector field X on M , (dF )X is awell-defined vector field on N . For any f ∈ C∞(N), the chain rule in local coordinates∂yj

∂xi(p) ∂

∂yj

∣∣F (p)

f = ∂∂xi

∣∣p(f F ) (xi on M and yj on N) yields a coordinate-free relation(

((dF )X)f) F = X(f F ), (1.10)

Let X, Y be vector fields regarded as differential operators on C∞(M). Then [X, Y ] =XY−Y X defines a vector field: its local expression is (Xj

∂Yi

∂xj−Yj

∂Xi

∂xj) ∂

∂xi. Direct calculation

shows that [·, ·] satisfies the Jacobi identity and so the space V (M) of all (smooth) vectorfields on a manifold is an infinite-dimensional Lie algebra.

Left-invariant vector fields

Let G be a Lie group, e ∈ G the identity element, and denote g = TeG. The groupoperations can be used to construct non-vanishing vector fields on G as follows.

For every g ∈ G, the left translation Lg : h ∈ G → gh ∈ G is a diffeomorphism of G.Let ξ ∈ g be a non-zero element. Define

Xξ : g ∈ G→ (dLg)eξ ∈ TgG ⊂ TG. (1.11)


Then Xξ 6= 0 at any point g ∈ G, for any ξ 6= 0, because the linear map (dLg)e is invertible.Furthermore, Xξ is a well-defined smooth vector field on G.

To verify the latter claim we consider the smooth map L : (g, h) ∈ G × G → gh ∈ G,using local coordinate charts ϕg0 : Ug0 → Rm, ϕe : Ue → Rm defined near g0, e ∈ G,respectively. Here m = dimG. The local expression of L near (g0, e) via these charts,Lloc = ϕg0 L(ϕ−1

g0, ϕ−1

e ), is a smooth map Lloc : Ug0×Ue → Ug0 . Now the local expressionfor (dLg)e is just the partial derivative map D2Lloc, linearizing Lloc in the second m-tupleof variables. It is clearly smooth in the first m variables, and so (dLg)eξ depends smoothlyon g.

To sum up, we have proved

Proposition 1.12. If ξ1, . . . , ξm is a basis of the vector space g then Xξ1(h), . . . , Xξm(h)define m = dimG vector fields whose values at each h ∈ G give a basis of ThG.

Hence, in view of Theorem 1.9, we obtain

Theorem 1.13. The tangent bundle TG of any Lie group G is isomorphic to the productG× Rdim G.

The smoothness of Xξ and (1.11) together imply that (dLg)hXξ(h) = Xξ(gh), i.e.

(dLg)Xξ = Xξ Lg. (1.14)

and we shall call any vector field satisfying (1.14) left-invariant.It follows from Proposition 1.12 that the vector space l(G) of all the left-invariant vector

fields on G form a finite-dimensional subspace (of dimension m) in the space V (G) of allvector fields. In fact more is true.

Theorem 1.15. The space of all the left-invariant vector fields on G is a finite-dimensionalLie algebra, hence a Lie subalgebra of V (G).

Proof. The theorem may be restated as saying that for every pair Xξ, Xη of left-invariantvector fields, [Xξ, Xη] is again left-invariant. To this end, we calculate using (1.10)

(dLg)[Xξ, Xη]f Lg = [Xξ, Xη](f Lg) = XξXη(f Lg)−XηXξ(f Lg)

= Xξ

((dLg)Xηf Lg

)−Xη

((dLg)Xξf Lg

)= (dLg)Xξ((dLg)Xηf) Lg − (dLg)Xη((dLg)Xξf) Lg

= [(dLg)Xξ, (dLg)Xη]f Lg

and using the left-invariant property (1.14) of Xξ and Xη for the next step

= [Xξ Lg, Xη Lg]f Lg = ([Xξ, Xη] Lg)f Lg.

Thus (dLg)[Xξ, Xη]f Lg = ([Xξ, Xη]Lg)f Lg, for each g ∈ G, f ∈ C∞(G), so the vectorfield [Xξ, Xη] satisfies (1.14) and is left-invariant.

It follows that [Xξ, Xη] = Xζ for some ζ ∈ g, thus the Lie bracket on l(G) induces oneon g. We can identify this Lie bracket more explicitly for the matrix Lie groups.

Theorem 1.16. If G is a matrix Lie group, then the map ξ ∈ g → Xξ ∈ l(G) is anisomorphism of the Lie algebras, where the Lie bracket on g is as defined in Theorem 1.7.

We shall prove Theorem 1.16 in the next section.

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1.4 Submanifolds

Suppose that M is a manifold, N ⊂ M , and N is itself a manifold, denote by ι : N → Mthe inclusion map.

Definition. N is said to be an embedded submanifold8 of M if(i) the map ι is smooth;(ii) the differential (dι)p at any point of p ∈ N is an injective linear map;(iii) ι is a homeomorphism onto its image, i.e. a D ⊆ N is open in the topology of

manifold N if and only if D is open in the topology induced on N from M (i.e. the opensubsets in N are precisely the intersections with N of the open subsets in M).

Remark . Often the manifold N is not given as a subset of M but can be identified with asubset of M by means of an injective map ψ : N →M . In this situation, the conditions inthe above definition make sense for ψ(N) (regarded as a manifold diffeomorphic to N). Ifthese conditions hold for ψ(N) then one says that the map ψ embeds N in M and writesψ : N →M .

Example. A basic example of embedded submanifold is a (parameterized) curve or surfacein R3. Then the condition (i) means that the parameterization is smooth and (ii) meansthat the parameterization is regular (recall introductory remarks on p.1). The condition(iii) eliminates e.g. the irrational twist flow t ∈ R → (t, αt) ∈ R2/Z2 = S1×S1, α ∈ R\Q,on the torus.

A surface or curve in R3 (or more generally in Rn) is often defined by an equation ora system of equations, i.e. as the zero locus of a smooth map on R3 (respectively on Rn).E.g. x2 + y2− 1 = 0 (a circle) or (x2 + y2 + b2 + z2− a2)2− 4b2(x2 + y2) = 0, (a torus withradii b > a > 0). However a smooth (even polynomial) map may in general have ‘bad’points in its zero locus (Example sheet 1, Q.6, 8 and 9). When does a system of equationson a manifold define a submanifold?

Definition. A value q ∈ N of a smooth map f between manifolds M and N is called aregular value if for any p ∈ M such that f(p) = q the differential of f at p is surjective,(df)p(TpM) = TqN .

Theorem 1.17. Let f : M → N be a smooth map between manifolds and q ∈ N a regularvalue of f . The inverse image of a regular value P = f−1(q) = p ∈ M : f(p) = q(if it is non-empty) is an embedded submanifold of M , of dimension dimM − dimN .

We shall need the following result from advanced calculus.

Inverse Mapping Theorem. Suppose that f : U ⊆ Rn → Rn is a smooth map definedon an open set U , 0 ∈ U and f(0) = 0. Then f has an smooth inverse g, defined on someneighbourhood of 0 with g(0) = 0, if and only if (df)0 is an invertible linear map of Rn.

Note that the Inverse Mapping Theorem, as stated above, is a local result, valid only ifone restricts attention to a suitably chosen neighbourhood of a point. The statement willin general no longer hold with neighbourhoods replaced by manifolds. (Consider e.g. themap of R to the unit circle S1 ⊂ C given by f(x) = eix.)

8In these notes I will sometimes write ‘submanifold’ meaning ‘embedded submanifold’.


Proof of Theorem 1.17. Firstly, P is Hausdorff and second countable because M is so.Let p be an arbitrary point of P . We may assume without loss of generality that

there are local coordinates xi, yj, i = 1, . . . , n, j = 1, . . . , k = dimN , n + k = dimM ,defined in a neighbourhood of p in M and local coordinates defined in a neighbourhoodof f(p) in N such that xi(p) = yj(p) = 0 and f is expressed in these local coordinatesas a map f = (f1, . . . , fk) on a neighbourhood of 0 in Rn with values in Rk, f(0) = 0,det(∂fi/∂yj)(0, 0) 6= 0.

Then

det

(1 0

(∂fi/∂xi) (∂fi/∂yj)

)(0, 0) 6= 0

and so, by the Inverse Function Theorem, the xi, fj form a valid set of new local coordinateson a (perhaps smaller) neighbourhood of p in M . The local equation for the intersection ofP with that neighbourhood takes in the new coordinates a simple form fj = 0, j = 1, . . . , k.Furthermore the first projection (xi, fj) 7→ (xi) restricts to a homeomorphism from aneighbourhood of p in P onto a neighbourhood of zero in Rn. Define this first projectionto be a coordinate chart on P with xi the local coordinates. Then the family of all suchcharts covers P and it remains to verify that any two charts defined in this way are in factcompatible.

So let xi, fj, x′i, f

′j be two sets of local coordinates near p as above. Then, by the

construction, for every p in P , we have that x′i = x′i(x, f), f ′j = f ′j(x, f) with f ′j(x, 0) = 0identically in x. Therefore, (∂f ′i/∂xj)(0, 0) = 0. Then

det

(∂x′i′/∂xi ∂x′i′/∂fj

∂f ′j′/∂xi ∂f ′j′/∂fj

)(0, 0) = det

(∂x′i′/∂xi ∂x′i′/∂fj

0 ∂f ′j′/∂fj

)(0, 0) 6= 0,

so we must have det(∂x′i′/∂xj)(0) 6= 0 for the n× n Jacobian matrix, and the change fromxi’s to x′i′ ’s is a diffeomorphism near 0.

Remarks. 1. It is not true that every submanifold of M is obtainable as the inverse imageof a regular value for some smooth map on M . One counterexample is RP 1 → RP 2

(check it!).2. Sometimes in the literature one encounters a statement ‘a subset P is (or is not) a

submanifold of M ’. Every subset P of a manifold M has a topology induced from M . Itturns out that there is at most one smooth structure on the topological space P such thatP is an embedded submanifold of M , but I shall not prove it here.

Theorem 1.18 (Whitney embedding theorem). Every smooth n-dimensional manifold canbe embedded in R2n (i.e. is diffeomorphic to a submanifold of R2n).

We shall assume the Whitney embedding theorem without proof here (a proof of em-bedding in R2n+1 is found in e.g. Guillemin and Pollack, Ch.1 §9).

It is worth to remark that the possibility of embedding any manifold in some RN , withN possibly very large, does not particularly simplify the study of manifolds in practicalterms (but is relatively easier to prove). The essence of the Whitney embedding theorem is

12 alexei kovalev

the minimum possible dimension of the ambient Euclidean space as way of measuring the‘topological complexity’ of the manifold. The result is sharp, in that the dimension of theambient Euclidean space could not in general be lowered (as can be checked by consideringe.g. the Klein bottle).

We can now give, as promised, a proof of Theorem 1.16

Theorem 1.16. Suppose that G ⊂ GL(n,R) is a subgroup and an embedded submani-fold of GL(n,R), and smooth structure on G is defined by the log-charts. Then the mapξ ∈ g → Xξ ∈ l(G) is an isomorphism of the Lie algebras, where the Lie bracket on g isthe Lie bracket of matrices, as in Theorem 1.7.

Proof of Theorem 1.16. We want to show [Xξ, Xη] = X[ξ,η] for a matrix Lie group wherethe LHS is the Lie bracket of left-invariant vector fields and the RHS is defined usingTheorem 1.7. Note first that for G = GL(nR), all the calculations can be done on an opensubset of Rn2

= Matr(n,R) with coordinates xij, i.j = 1. . . . , n. The map Lg, and hence

also dLg, is the usual left multiplication by a fixed matrix g = (xij). Respectively, the

left-invariant vector fields are Xξ(g) = xikξ

kj

∂∂xi

jand [Xξ, Xη] = X[ξ,η] is an easy calculation.

Thus the theorem holds for GL(n,R).Now consider a general case G ⊂ GL(n,R). Denote, as before, the inclusion map by ι.

Any left-invariant vector field Xξ on G (ξ ∈ g) may be identified by means of dι with avector field defined on a subset G ⊂ GL(n,R). Further, the left translation Lg on G isthe restriction of the left translation on GL(n,R) (if g ∈ G). We find that the vectorfields (dι)Xξ (ξ ∈ g) correspond bijectively to the restrictions to G ⊂ GL(n,R) of theleft-invariant vector fields Xξ on GL(n,R), such that ξ ∈ g. Let Xξ, Xη ∈ l(GL(n,R))with ξ, η ∈ g, where g is understood as the image of log-chart for G near I. We have[Xξ, Xη] = X[ξ,η], by the above calculations on GL(n,R), and the Lie bracket of matrices[ξ, η] ∈ g by Theorem 1.7. Therefore, X[ξ,η] restricts to give a well-defined left-invariantvector field on G, and [Xξ, Xη] = X[ξ,η] holds for any Xξ, Xη ∈ l(G) as claimed.

Remark. Notice that the differential dι considered in the above proof identifies g with alinear subspace of n× n matrices, by considering G as a hypersurface in GL(n,R) ⊂ Rn2

.In the example of G = O(n), for any path A(t) or orthogonal matrices with A(0) = I,we calculate 0 = d

dt

∣∣t=0A(t)A∗(t) = A(0)A∗(0) + A(0)A∗(0) = A(0) + A∗(0), i.e. A(0) is

skew-symmetric. Thus the log chart at the identity actually maps onto a neighbourhoodof zero in the tangent space TIG —which explains why the log-chart construction in §1.2worked.

1.5 Exterior algebra of differential forms. De Rham cohomology

See the reference card on multilinear algebra

The differential forms

Consider a smooth manifold M of dimension n. The dual space to the tangent space TpM ,p ∈M , is called the cotangent space to M at p, denoted T ∗pM . Suppose that a chart and

hence the local coordinates are given on a neighbourhood of p. The dual basis to ( ∂∂xi

)p is


traditionally denoted by (dxi)p, i = 1, . . . , n. (Sometimes I may drop the subscript p fromthe notation.) Thus an arbitrary element of T ∗pM is expressed as

∑i ai(dxi)p for some

ai ∈ R.Recall from (1.4′) that a change of local coordinates, say from from xi to x′i, induces

a change of basis of tangent space TpM from ∂∂xi

to ∂∂x′

i. By linear algebra, there is a

corresponding change of the dual basis of T ∗pM , from dx′i to dxi given by the transposedmatrix. Thus the transformation law is

dx′j =∂x′j∂xi

dxi. (1.19)

Like (1.4′) the eq.(1.19) is a priori merely a notation which resembles familiar results fromthe calculus. See however further justification in the remark on the next page.

A disjoint union of all the cotangent spaces T ∗M = tp∈MT∗pM of a given manifold

M is called the cotangent bundle of M . The cotangent bundle can be given a smoothstructure making it into a manifold of dimension 2 dimM by an argument very similarto one for the tangent bundle (but with a change of notation, replacing any occurrenceof (1.4′) with (1.19)).

A smooth field of linear functionals is called a (smooth) differential 1-form (or just1-form). More precisely, a differential 1-form is a map α : M → T ∗M such that αp ∈ T ∗pMnear every p ∈ M and α is expressed in any local coordinates x = (x1, . . . , xn) by α =∑

i ai(x)dxi where ai(x) are some smooth functions of x.

Remark . The 1-forms are of course the dual objects to the vector fields. In particular,ai(x(p)) is obtained as the value of αp on the tangent vector ( ∂

∂xi)p. Consequently, α is

smooth on M if and only if α(X) is a smooth function for every vector field X on M .Notice that the latter condition does not use local coordinates.

One can similarly consider a space ΛrT ∗pM of alternating multilinear functions on TpM×. . .×TpM(r factors), for any r = 0, 1, 2, . . . n, and proceed to define the r-th exterior powerΛrT ∗M of the cotangent bundle of M and the (smooth) differential r-forms on M .Details are left as an exercise.

The space of all the smooth differential r-forms on M is denoted by Ωr(M) and r isreferred to as the degree of a differential form. If r = 0 then Λ0T ∗M = M × R andΩ0(M) = C∞(M). The other extreme case r = dimM is more interesting.

Theorem 1.20 (Orientation of a manifold). Let M be an n-dimensional manifold. Thefollowing are equivalent:

(a) there exists a nowhere vanishing smooth differential n-form on M ;(b) there exists a family of charts in the differentiable structure on M such that the re-

spective coordinate domains cover M and the Jacobian matrices have positive determinantson every overlap of the coordinate domains;

(c) the bundle of n-forms ΛnT ∗M is isomorphic to M × R.

Proof (gist). That (a)⇔(c) is proved similarly to the proof of Theorem 1.9.

14 alexei kovalev

Using linear algebra, we find that the transformation of the differential forms of topdegree under a change of coordinates is given by

dx1 ∧ . . . ∧ dxn = det(∂xj

∂x′i

)dx′1 ∧ . . . ∧ dx′n.

Now (a)⇒(b) is easy to see.To obtain, (b)⇒(a) we assume the following.

Theorem 1.21 (Partition of unity). For any open cover M ⊂ ∪α∈AUα, there exists acountable collection of functions ρi ∈ C∞(M), i = 1, 2, ..., such that the following holds:(i) for any i, the closure of supp(ρi) = x ∈ M : ρi(x) 6= 0 is compact and contained inUα for some α = αi (i.e. depending on i);(ii) the collection is locally finite: each x ∈M has a neighbourhood Wx such that ρi(x) 6= 0on Wx for only finitely many i; and(iii) ρi ≥ 0 on M for all i and

∑i ρi(x) = 1 for all x ∈M .

The collection ρi satisfying the above is called a partition of unity subordinate to Uα.

Choose a partition of unity ρi subordinate to the given family of coordinate neigh-

bourhoods covering M . For each i, choose local coordinates x(α)i valid on the support of ρi.

Define ωi = dx(α)1 ∧. . .∧dx(α)

n in these local coordinates, then ραωα is a well-defined (smooth)n-form on all of M (extended by zero outside the coordinate domain) and ω =

∑α ραωα

is the required n-form.

A manifold M satisfying any of the conditions (a),(b),(c) of the above theorem iscalled orientable. A choice of the differential form in (a), or family of charts in (b),or diffeomorphism in (c) defines an orientation of M and a manifold endowed with anorientation is said to be oriented.

Exterior derivative

Recall that the differential of a smooth map f : M → N between manifolds is a linear mapbetween respective tangent spaces. In the special case N = R, the (df)p at each p ∈ M isa linear functional on TpM , i.e. an element of the dual space T ∗pM . In local coordinates

xi defined near p we have df(x) = ∂f∂xi

(x)dxi, thus df is a well-defined differential 1-form,whose coefficients are those of the gradient of f .

Remark . Observe that any local coordinate xi on an open domain U ⊂ M is a smoothfunction on U . Then the formal symbols dxi actually make sense as the differentials ofthese smooth functions (which justifies the previously introduced notation, cf.(1.19)).

Theorem 1.22 (exterior differentiation). There exists unique linear operatord : Ωk(M) → Ωk+1(M), k ≥ 0, such that

(i) if f ∈ Ω0(M) then df coincides with the differential of a smooth function f ;

(ii) d(ω ∧ η) = dω ∧ η + (−1)deg ωω ∧ dη for any two differential forms ω,η;

(iii) ddω = 0 for every differential form ω.


Proof (gist). On an open set U ⊆ Rn, or in the local coordinates on a coordinate domainon a manifold, application of conditions (ii), then (iii) and (i), yields

d(fdxi1 ∧ . . . ∧ dxir) = df ∧ dxi1 ∧ . . . ∧ dxir =∂f

∂xi

dxi ∧ dxi1 ∧ . . . ∧ dxir , (1.23)

for any smooth function f . Extend this to arbitrary differential forms by linearity. Theconditions (i),(ii),(iii) then follow by direct calculation, in particular the last of these holdsby independence of the order of differentiation in second partial derivatives. This provesthe uniqueness, i.e. that if d exists then it must be expressed by (1.23) in local coordinates.

Observe another important consequence of (1.23): the operator d is necessarily local,which means that the value (dω)p at a point p is determined by the values of differentialform ω on a neighbourhood of p.

To establish the existence of d one now needs to show that the defining formula (1.23)is consistent, i.e. the result of calculation does not depend on the system of local coordi-nates in which it is performed. So let d′ denote the exterior differentiation constructed asin (1.23), but using different choice of local coordinates. Then, by (ii), we must have

d′(fdxi1 ∧ . . . ∧ dxir) = d′f ∧ dxi1 ∧ . . . ∧ dxir +r∑

j=1

(−1)jfdxi1 ∧ . . . d′(dxij) . . . ∧ dxir .

But d′f = df and d′(dxk) = d′(d′xk) = 0, by (i) and (iii) and because we know thatthe differential of a smooth function (0-form) is independent of the coordinates. Hence theright-hand side of the above equality becomes ∂f

∂xjdxj∧dxi1∧. . .∧dxir = d(fxi1∧. . .∧dxir),

and so the exterior differentiation is well-defined.

De Rham cohomology

A differential form α is said to be closed when dα = 0 and exact when α = dβ for somedifferential form β. Thus exact forms are necessarily closed (but the converse is not ingeneral true, e.g. example sheet 2, Q4).

Definition. The quotient space

HkdR(M) =

closed k-forms on M

exact k-forms on M

is called the k-th de Rham cohomology group of the manifold M .

Any smooth map between manifolds, say f : M → N , induces a pull-back mapbetween the cotangent spaces f ∗ : T ∗f(p)N → T ∗pM, which is a linear map defined, for anydifferential form α on N , by

(f ∗α)p(v1, . . . , vr) = αf(p)((df)pv1, . . . , (df)pvr),

16 alexei kovalev

using the differential of f . The chain rule for differentials of smooth maps immediatelygives

(f g)∗ = g∗f ∗. (1.24)

It is also straightforward to check that f ∗ preserves the ∧-product f ∗(α∧β) = (f ∗α)∧(f ∗β)and f ∗ commutes with the exterior differentiation, f ∗(dα) = d(f ∗α), hence f ∗ preserves thesubspaces of closed and exact differential forms. Therefore, every smooth map f : M → Ninduces a linear map on the de Rham cohomology

f ∗ : Hr(N) → Hr(M)

A consequence of the chain rule (1.24) is that if f is a diffeomorphism then f ∗ is a linearisomorphism. Thus the de Rham cohomology is a diffeomorphism invariant, i.e. diffeo-morphic manifolds have isomorphic de Rham cohomology.9

Poincare lemma. Hk(D) = 0 for any k > 0, where D denotes the open unit ball in Rn.

The proof goes by working out a way to invert the exterior derivative. More precisely,one constructs linear maps hk : Ωk(U) → Ωk−1(U) such that

hk+1 d+ d hk = idΩk(U) .

Remark . In the degree 0, one has H0(M) = R for any connected manifold.

Basic integration on manifolds

Throughout this subsection M is an oriented n-dimensional manifold. Let ω ∈ Ωn(M)and, as before, denote suppω = p ∈ M : ωp 6= 0 (the support of ω). Suppose that theclosure of suppω is compact.

Consider first the special case when the closure of suppω is contained in the domainof just one coordinate chart, (U,ϕ) say. If f(x)dx1 ∧ . . . ∧ dxn is the local expressionfor ω then the integral

∫ϕ(Uα)

f(x)dx1 . . . dxn makes sense as in the multivariate calculus.

The value of this integral is independent of the choice of local coordinates, provided onlythat the change is orientation-preserving, i.e. the Jacobian is positive. This is because thelocal expression for ω changes precisely as required by the change of variables formula forintegrals of functions of n variables (which involves the absolute value of the Jacobian).Thus

∫Mω =

∫Uω is well defined when ω supported in just one coordinate chart.

Now let ω be any n-form with compact support. Consider an oriented system of charts(Uα, ϕα) covering M (as in Theorem 1.20(b)). Let (ρα) be a partition of unity subordinateto Uα.

Definition. In the above situation, the integral of ω over M is given by∫M

ω =∑

α

∫Uα

ραω.

9In fact, more is true. It can be shown, using topology, that the de Rham cohomology depends only onthe topological space underlying a smooth manifold and that homeomorphic manifolds have isomorphic deRham cohomology. The converse in not true: there are manifolds with isomorphic de Rham cohomologybut e.g. with different fundamental groups.


Note that the sum in the right-hand side is finite. It can be checked that∫

Mω does

not depend on the choice of a partition of unity on M , and is therefore well-defined.

Stokes’ Theorem (for manifolds without boundary)10. Suppose that η ∈ Ωn−1(M)has a compact support. Then

∫Mdη = 0.

Let ρα be a partition of unity subordinate to some oriented system of coordinate neigh-bourhoods covering M (as above). Then dη is a finite sum of the forms ραη. Now the proofof Stokes’ Theorem can be completed by considering compactly supported exact forms onRn and using calculus.

Corollary 1.25 (Integration by parts). Suppose that α and β are compactly supporteddifferential forms on M and degα+ deg β = dimM − 1.

Then∫

Mα ∧ dβ = (−1)1+deg α

∫M

(dα) ∧ β.

One rather elegant application of the results discussed above is the following.

Theorem 1.26. Every (smooth) vector field on S2n vanishes at some point.

Proof of Theorem 1.26. Notation: recall that we write Sn ⊂ Rn+1 for the unit sphere aboutthe origin. For r > 0, let ι(r) : x ∈ Sn → rx ∈ Rn+1 denote the embedding of Sn in theEuclidean space as the sphere of radius r about the origin and write Sn(r) = ι(r)(Sn)(thus, in particular, Sn(1) = Sn).

Suppose, for a contradiction, that X(x) is a nowhere-zero vector field on Sn. Wemay assume, without loss of generality, that |X| = 1, identically on Sn. For any realparameter ε, define a map

f : x ∈ Rn+1 \ 0 → x+ ε|x|X(x/|x|) ∈ Rn+1 \ 0.

Here we used the inclusion Sn ⊂ Rn+1 (and hence TpSn ⊂ TpRn+1 ∼= Rn+1) to define a

(smooth) map x ∈ Rn+1 \ 0 → X(x/|x|) ∈ Rn+1 \ 0.

Step 1.We claim that f is an diffeomorphism, whenever |ε| is sufficiently small. Firstly, for anyx0 6= 0,

(df)x0 = idRn+1 +ε[d(|x|X(x/|x|))

]x0

and straightforward calculus shows that the norm of the linear map defined by the Jacobimatrix

[d(|x|X(x/|x|))

]x0

is bounded independent of x0 6= 0. Hence there is ε0 > 0, such

that (df)x0 is a linear isomorphism of Rn+1 onto itself for any x0 6= 0 and any |ε| < ε0. Butthen, by the Inverse Mapping Theorem (page 10), for any x 6= 0, f maps some open ballB(x, δx) of radius δx > 0 about x diffeomorphically onto its image.

Furthermore, it can be checked, by inspection of the proof of the Inverse Mapping The-orem, that (1) δx can be taken to be continuous in x and (2) δx can be chosen independentof ε if |ε| < ε0. We shall assume these two latter claims without proof. Consequently, δxcan be taken to depend only on |x| (as Sn(|x|) is compact).

10Manifolds with boundary are not considered in these lectures.

18 alexei kovalev

Taking a smaller ε0 > 0 if necessary, we ensure that f is one-to-one if |ε| < ε0. Forwe have |f(x)| =

√1 + ε2|x| and so it suffices to check that that f is one-to-one on each

Sn(|x|). But two points on Sn(|x|) far away from each other cannot be mapped to onebecause |f(x) − x| < ε|x| and two distinct points at a distance less than say 1

2δ|x| cannot

be mapped to one because f restricts to a diffeomorphism (hence a bijection) on a δ|x|-ballabout each point.

A similar reasoning shows that f is surjective (onto) if ε in sufficiently small. Indeed,f(B(x, δx)) is an open set homeomorphic to a ball and the boundary of f(B(x, δx)) iswithin small distance ε(1 + δx)|x| from the boundary of B(x, δx). Therefore, x must beinside the boundary of f(B(x, δx)) and thus in the image of f . In all of the above, ‘smallε0’ can be chosen independent of x because δx depends only on |x| and f is homogeneousof degree 1, f(λx) = λf(x) for each positive λ.

p.t.o.


Step 2. Now, as f is a diffeomorphism f maps the embedded submanifold Sn(1)

diffeomorphically onto the embedded submanifold f(Sn(1)) = Sn(√

1 + ε2). Consider adifferential n-form on Rn+1

ω =n∑

i=0

(−1)ixidx0 ∧ . . . (omit dxi) . . . ∧ dxn,

We have∫

f(Sn(1))ω =

∫Sn(1)

f ∗ω (change of variable formula) and this integral depends

polynomially on the parameter ε because f ∗ω does so (as f is linear in ε).But on the other hand, for any r > 0,∫

Sn(r)

ω −∫

Sn(1)

ω =

∫Sn

ι(r)∗ω −∫

Sn

ω

=

∫ r

1

d

ds

(∫Sn

ι(s)∗ω

)ds

=

∫ r

1

(∫Sn

d

ds

(ι(s)∗ω

))ds

applying, on each coordinate patch, a theorem on differentiation of an integral dependingon a parameter s, from calculus

=

∫1≤|x|≤r

dω

replacing, again on each coordinate patch, a repeated integration with an (n+1)-dimensionalintegral

=

∫1≤|x|≤r

(n+ 1)dx0 ∧ . . . ∧ dxn = cn+1(rn+1 − 1),

where cn+1 is n + 1 times the volume of (n + 1)-dimensional ball (the value of cn+1 doesnot matter here). Put r =

√1 + ε2 and then the right-hand side is not a polynomial in ε

if n+ 1 is odd (i.e. when n is even). A contradiction.

Some page references

20 alexei kovalev

to Warner, and Guillemin–Pollack, and Gallot–Hulin–Lafontaine.N.B. The material in these books is sometimes covered differently from the Lectures andmay contain additional topics, thus the references are not quite ‘one-to-one’.

smooth manifolds [W] 1.2–1.6tangent and cotangent bundles [W] 1.25exponential map on a matrix Lie group [W] 3.35left invariant vector fields [W] 3.6–3.7submanifolds [W] 1.27–1.31, 1.38differential forms [GP] 153–165,174–178de Rham cohomology [GP] pp.178–182Poincare Lemma [W] 4.18partition of unity [GP] p.52 or [W] 1.8–1.11integration and Stokes’ Theorem [GP] 165–168, 183–185non-existence of vector fields without zeros on S2n (Milnor’s proof) [GHL] 1.41

Part III: Differential geometry (Michaelmas 2004)


2 Vector bundles.

Definition. Let B be a smooth manifold. A manifold E together with a smooth submer-sion1 π : E → B, onto B, is called a vector bundle of rank k over B if the followingholds:

(i) there is a k-dimensional vector space V , called typical fibre of E, such that for anypoint p ∈ B the fibre Ep = π−1(p) of π over p is a vector space isomorphic to V ;

(ii) any point p ∈ B has a neighbourhood U , such that there is a diffeomorphism

π−1(U)ΦU−−−→ U × V

π

y ypr1

U U

and the diagram commutes, which means that every fibre Ep is mapped to p × V .

ΦU is called a local trivialization of E over U and U is a trivializing neighbour-hood for E.

(iii) ΦU |Ep : Ep → V is an isomorphism of vector spaces.

Some more terminology: B is called the base and E the total space of this vector bundle.

π : E → B is said to be a real or complex vector bundle corresponding to the typical fibrebeing a real or complex vector space. Of course, the linear isomorphisms etc. are understoodto be over the respective field R or C. In what follows vector bundles are taken to be realvector bundles unless stated otherwise.

Definition. Any smooth map s : B → E such that π s = idB is called a section of E.If s is only defined over a neighbourhood in B it is called a local section.

Examples. 0. A trivial, or product, bundle E = B × V with π the first projection.Sections of this bundle are just the smooth maps C∞(B;V ).

1. The tangent bundle TM of a smooth manifold M has already been discussed inChapter 1. It is a real vector bundle of rank n = dimM which in general is not trivial.2

The sections of TM are the vector fields. In a similar way, the cotangent bundle T ∗M and,

1A smooth map is called a submersion if its differential is surjective at each point.2Theorems 1.9 and 1.26 in Chapter 1 imply that TS2n cannot be trivial.

20


more generally, the bundle of differential p-forms ΛpT ∗M are real vector bundles of rank(np

)with sections being the differential 1-forms, respectively p-forms. Exercise: verify that

the vector bundles ΛpT ∗M (1 ≤ p ≤ dimM) will be trivial if TM is so.2. ‘Tautological vector bundles’ may be defined over projective spaces RP n, CP n (and,

more generally, over the Grassmannians). Let B = CP n say. Then let E be the disjointunion of complex lines through the origin in Cn+1, with π assigning to a point in p ∈ Ethe line ` containing that point, so π(p) = ` ∈ CP n. We shall take a closer look at oneexample (Hopf bundle) below and show that the tautological construction indeed gives awell-defined (and non-trivial) complex vector bundle of rank 1 over CP 1.

Structure group of a vector bundle.

It follows from the definition of a vector bundle E that one can define over the intersectionof two trivializing neighbourhoods Uβ, Uα a composite map

Φβ Φ−1α (b, v) = (b, ψβα(b)v),

(b, v) ∈ (Uβ ∩ Uα) × Rk. For every fixed b the above composition is a linear isomorphismof Rk depending smoothly on b. The maps ψβα : Uβ ∩ Uα → GL(k,R). are called thetransition functions of E.

It is not difficult to see that transition functions ψαβ satisfy the following relations,called ‘cocycle conditions’

ψαα = idRk ,ψαβψβα = idRk ,ψαβψβγψγα = idRk .

(2.1)

The left-hand side is defined on the intersection Uα ∩ Uβ, for the second of the aboveequalities, and on Uα ∩ Uβ ∩ Uγ for the third. (Sometimes the name ‘cocycle condition’refers to just the last of the equalities (2.1); the first two may be viewed as notation.)

Now it may happen that a vector bundle π : E → B is endowed with a system of trivi-alizing neighbourhoods Uα covering the base and such that all the corresponding transitionfunctions ψβα take values in a subgroup G ⊆ GL(k,R), ψβα(b) ∈ G for all b ∈ Uβ ∩ Uα, forall α, β, where k is the rank of E. Then this latter system (Uα,Φα) of local trivializationsover Uα’s is said to define a G-structure on vector bundle E.

Examples. 0. If G consists of just one element (the identity) then E has to be a trivialbundle E = B × Rk.

1. Let G = GL+(k,R) be the subgroup of matrices with positive determinant. If thetypical fibre Rk is considered as an oriented vector space then the transition functions ψβαpreserve the orientation. The vector bundle E is then said to be orientable.

A basic example arises from a system of coordinate charts giving an orientation of amanifold M . The transition functions of TM are just the Jacobians and so M is orientableprecisely when its tangent bundle is so.

2. A more interesting situation occurs when G = O(k), the subgroup of all the non-singular linear maps in GL(k,R) which preserve the Euclidean inner product on Rk. It

22 alexei kovalev

follows that the existence of an O(k) structure on a rank k vector bundle E is equivalentto a well-defined positive-definite inner product on the fibres Ep. This inner productis expressed in any trivialization over U ⊆ B as a symmetric positive-definite matrixdepending smoothly on a point in U .

Conversely, one can define a vector bundle with inner product by modifying the defi-nition on page 18: replace every occurrence of ‘vector space’ by ‘inner product space’ and(linear) ‘isomorphism’ by (linear) ‘isometry’. This will force all the transition functions totake value in O(k) (why?).

A variation on the theme: an orientable vector bundle with an inner product is thesame as vector bundle with an SO(k)-structure.

3. Another variant of the above: one can play the same game with rank k complexvector bundles and consider the U(k)-structures (U(k) ⊂ GL(k,C)). Equivalently, considercomplex vector bundles with Hermitian inner product ‘varying smoothly with the fibre’.Furthermore, complex vector bundles themselves may be regarded as rank 2k real vectorbundles with a GL(k,C)-structure (the latter is usually called a complex structure on avector bundle).

In the examples 2 and 3, if a trivialization Φ is ‘compatible’ with the given O(k)- orSO(k)-structure (respectively U(k)-structure) (Uα,Φα) in the sense that the transitionfunctions Φα Φ−1 take values in the orthogonal group (respectively, unitary group) thenΦ is called an orthogonal trivialization (resp. unitary trivialization).

Principal bundles.

Let G be a Lie group. A smooth free right action of G on a manifold P is a smoothmap P × G → P , (p, h) 7→ ph, such that (1) for any p ∈ P , ph = p if and only if h is theidentity element of G; and (2) (ph1)h2 = p(h1h2) for any p ∈ P , any h1, h2 ∈ G. (It followsthat for each h ∈ G, P × h → P is a diffeomorphism.)

Definition . A (smooth) principal G-bundle P over B is a smooth submersionπ : P → B onto a manifold B, together with a smooth right free action P ×G→ P , suchthat the set of orbits of G in P is identified with B (as a set), P/G = B, and also for anyb ∈ B there exists a neighbourhood U ⊆ B of b and a diffeomorphism ΦU : π−1(U) → U×Gsuch that pr1 ΦU = π|π−1(U), i.e. the following diagram is commutative

π−1(U)ΦU−−−→ U ×G

π

y ypr1

U U

(2.2)

and ΦU commutes with the action of G, i.e. for each h ∈ G, ΦU(ph) = (b, gh), where(b, g) = ΦU(p), π(p) = b ∈ U .

A local section of the principal bundle P is a smooth map s : U → P defined on aneighbourhood U ⊂ B and such that π s = idU .


For a pair of overlapping trivializing neighbourhoods Uα, Uβ one has

Φβ Φ−1α (b, g) = (b, ψβα(b, g)),

where for each b ∈ Uα ∩Uβ, the ψβα(b, ·) is a map G→ G. Then, for each b in the domainof ψβα, we must have ψβα(b, g)h = ψβα(b, gh) for all g, h ∈ G, in view of (2.2). It follows (bytaking g to be the unit element 1G) that the map ψβα(b, ·) is just the multiplication on theleft by ψβα(b, 1G) ∈ G. It is sensible to slightly simplify the notation and write g 7→ ψβα(b)gfor this left multiplication. We find that, just like vector bundles, the principal G-bundleshave transition functions ψβα : Uα ∩ Uβ → G between local trivializations. In particular,ψβα for a principal bundle satisfy the same cocycle conditions (2.1).

A principal G-bundle over B may be obtained from a system of ψβα, correspondingto an open cover of B and satisfying (2.1), — via the following ‘Steenrod construction’.For each trivializing neighbourhood Uα ⊂ B for E consider Uα×G. Define an equivalencerelation between elements (b, h) ∈ Uα×G, (b′, h′) ∈ Uβ×G, so that (b, h) ∼ (b′, h′) preciselyif b′ = b and h′ = ψβα(b)h. Now let

P = tα(Uα ×G

)/ ∼ (2.3)

the disjoint union of all Uα ×G’s glued together according to the equivalence relation.

Theorem 2.4. P defined by (2.3) is a principal G-bundle.

Remark . The ψβα’s can be taken from some vector bundle E over B, then P will be‘constructed from E’. The construction can be reversed, so as to start from a principalG-bundle P over a base manifold B and obtain the vector bundle E over B. Then E willbe automatically given a G-structure.

In either case the data of transition functions is the same for the principal G-bundle Pand the vector bundle P . The difference is in the action of the structure group G on thetypical fibre. G acts on itself by left translations in the case of the principal bundle and Gacts as a subgroup of GL(k,R) on Rk in the case of vector bundle E. 3 The vector bundleE is then said to be associated to P via the action of G on Rk.

Example: Hopf bundle.

Hopf bundle may be defined as the ‘tautological’ (see p.19) rank 1 complex vector bundleover CP 1. The total space E of Hopf bundle, as a set, is the disjoint union of all (complex)lines passing through the origin in C2. Recall that every such line is the fibre over thecorresponding point in CP 1. We shall verify that the Hopf bundle is well-defined byworking its transition functions, so that we can appeal to Theorem 2.4.

For a covering system of trivializing neighbourhoods in CP 1, we can choose the coor-dinate patches of the smooth structure of CP 1, defined in Chapter 1. Thus

CP 1 = U1 ∪ U2, Ui = z1 : z2 ∈ CP 1, zi 6= 0, i = 1, 2,

3G need not be explicitly a subgroup of GL(k, R), it suffices to have a representation of G on Rk.

24 alexei kovalev

with the local complex coordinate z = z2/z1 on U1, and ζ = z1/z2 on U2, and ζ = 1/z whenz 6= 0. We shall denote points in the total space E as (wz1, wz2), with |z1|2 + |z2|2 6= 0,w ∈ C, so as to present each point as a vector with coordinate w relative to a basis (z1, z2)of a fibre. (This clarification is needed in the case when (wz1, wz2) = (0, 0) ∈ C2.) An‘obvious’ local trivialization over Ui may be given, say over U1, by (w,wz) ∈ π1(U1) →(1 : z, w) ∈ U1 × C, —but in fact this is not a very good choice. Instead we define

Φ1 : (w,wz) ∈ π1(U1) → (1 : z, w√

1 + |z|2) ∈ U1 × C,

Φ2 : (wζ,w) ∈ π1(U2) → (ζ : 1, w√

1 + |ζ|2) ∈ U2 × C.

Calculating the inverse, we find

Φ−11 (1 : z, w) =

(w√

1 + |z|2,

wz√1 + |z|2

)and so

Φ2 Φ−11 (1 : z, w) = Φ2

(w√

1 + |z|2,

wz√1 + |z|2

)= Φ2

(|ζ|w

ζ√|ζ|2 + 1

ζ,|ζ|w

ζ√|ζ|2 + 1

)=

(ζ : 1,

|ζ|ζw

)=

(1 : z,

z

|z|w

)giving the transition function τ2,1(1 : z) = (z/|z|), for 1 : z ∈ U1 ∩ U2 (i.e. z 6= 0). Theτ2,1 takes values in the unitary group U(1) = S1 = z ∈ C : |z| = 1, a subgroup ofGL(1,C) = C \ 0 (it is for this reason the square root factor was useful in the localtrivialization). Theorem 2.4 now ensures that Hopf bundle E is a well-defined vectorbundle, moreover a vector bundle with a U(1)-structure. Hence there is a invariantlydefined notion of length of any vector in any fibre of E. The length of (wz1, wz2) maybe calculated in the local trivializations Φ1 or Φ2 by taking the mudulus of the secondcomponent of Φi (in C). For each i = 1, 2, this coincides with the familiar Hermitianlength

√|wz1|2 + |wz2|2 of (wz1, wz2) in C2.

We can now use τ2,1 to construct the principal S1-bundle (i.e. U(1)-bundle) P → CP 1

associated to Hopf vector bundle E, cf. Theorem 2.4. If U(1) is identified with a unitcircle S1 ⊂ C , any fibre of P may be considered as the unit circle in the respectivefibre of E. Thus P is identified as the space of all vectors in E of length 1, so P =(w1, w2) ∈ C2 | w1w1+w2w2 =1 is the 3-dimensional sphere and the bundle projection is

π : (w1, w2) ∈ S3 → w1 : w2 ∈ S2, π−1(p) ∼= S1,

where we used the diffeomorphism S2 ∼= CP 1 for the target space. (Examples 1, Q3(ii).)This principal S1-bundle S3 over S2 is also called Hopf bundle. It is certainly not trivial,as S3 is not diffeomorphic to S2 × S1. (The latter claim is not difficult to verify, e.g.by showing that the de Rham cohomology H1(S3) is trivial, whereas H1(S2 × S1) is not.Cf. Examples 2 Q5.)


Pulling back vector bundles and principal bundles

Let P be a principal bundle over a base manifold B and E an associated vector bundleover B. Consider a smooth map f : M → B.

The pull-back of a vector (respectively, principal) bundle is a bundle f ∗E (f ∗P )over M such that there is a commutative diagram (vertical arrows are the bundle projec-tions)

f ∗EF−−−→ Ey y

Mf−−−→ B,

(2.5)

such that the restriction of F to each fibre (f ∗E)p over p ∈ M is an isomorphism onto afibre Ef(p).

A very basic special case of the above is when f maps M to a point in B; then the pull-back f ∗E (and f ∗P ) is necessarily a trivial bundle (exercise: write out a trivialization mapf ∗E → M × (typical fibre)). As a slight generalization of this example consider the casewhen M = B×X, for some manifold X with f : B×X → B the first projection. Then f ∗E(resp. f ∗P ) may be thought of as bundles ‘trivial in the X direction’, e.g. f ∗E ∼= E ×X,with the projection (e, x) ∈ E ×X → (π(e), x) ∈ B ×X.

The construction may be extended to a general vector bundle, by working in localtrivialization. Then one has to ensure that the pull-back must be well-defined independentof the choice of local trivialization. To this end, let ψβα be a system of transitionfunctions for E. Define

f ∗ψβα = ψβα f

and f ∗ψβα is a system of functions on M satisfying the cocycle condition (2.1). Therefore,by Theorem 2.4 and a remark following this theorem, the f ∗ψβα are transition functionsfor a well-defined vector bundle and principal bundle over M . Steenrod construction showsthat these are indeed the pull-back bundles f ∗E and f ∗P as required by (2.5).

26 alexei kovalev

2.1 Bundle morphisms and automorphisms.

Let (E,B, π) and (E ′, B′, π′) be two vector bundles, and f : B → B′ a smooth map.

Definition. A smooth map F : E → E ′ is a vector bundle morphism covering f iffor any p ∈ B F restricts to a linear map between the fibres F : Ep → E ′

f(p) for any p ∈ B,so that

EF−−−→ E ′

π

y π′

yB

f−−−→ B′

is a commutative diagram, π′ F = f π.

More explicitly, suppose that the local trivializations Φ : π−1(U) → U × V , Φ′ :π′−1(U ′) → U ′×V ′ are such that f(U) ⊆ U ′. Then the restriction FU = Φ′ F |π−1(U) Φ−1

is expressed asFU : (b, v) 7→ (f(b), h(b)v), (2.6)

for some smooth h : U → L(V, V ′) family of linear maps between vector spaces V ,V ′

depending on a point in the base manifold. In particular, it is easily checked that acomposition of bundle morphisms E → E ′, E ′ → E ′′ is a bundle morphism E → E ′′.

Examples. 1. If ϕ : M → N is a smooth map between manifolds M ,N then its differentialdϕ : TM → TN is a morphism of tangent bundles.

2. Recall the pull-back of a given vector bundle (E,B, π) via a smooth map f : M → B.For every local trivialization E|U of E, the corresponding local trivialization of the pull-back bundle is given by f ∗E|f−1(U) → (f−1(U))×V , where V is the typical fibre of E (hencealso of f ∗E). Thus the pull-back construction gives a well-defined map F : f ∗E → E whichrestricts to a linear isomorphism between any pair of fibres (f ∗E)p and Ef(p), p ∈M . (Thisisomorphism becomes just the identity map of the typical fibre V in the indicated localtrivializations.) It follows that F is a bundle morphism covering the given map f : M → B.

3. Important special case of bundle morphisms occurs when f is a diffeomorphism of Bonto B′. A morphism F : E → E ′ between two vector bundles over B covering f is calledan isomorphism of vector bundles if F restricts to a linear isomorphism Ep → Ef(p),for every fibre of E.

An isomorphism from a vector bundle E to itself covering the identity map idB iscalled a bundle automorphism of E. The set AutE of all the bundle automorphismsof E forms a group (by composition of maps). If E = B × V is a trivial bundle then anyautomorphism of E is defined by a smooth maps B → GL(V ), so AutE = C∞(B,GL(V )).

If a vector bundle E has a G-structure (G ⊆ GL(V )) then it is natural to consider thegroup of G-bundle automorphisms of E, denoted AutGE and defined as follows. Recallthat a G-structure means that there is a system of local trivializations over neighbourhoodscovering the base B and with the transition functions of E taking values inG. Now a bundleautomorphism F ∈ AutE of E → B is determined in any local trivialization over open


U ⊆ B by a smooth map h : U → GL(V ), as in (2.6). Call F a G-bundle automorphismif for any of the local trivializations defining the G-structure this map h takes values in thesubgroup G. It follows that AutGE is a subgroup of AutE. (In the case of trivial bundlesthe latter statement becomes C∞(B,G) ⊆ C∞(B,GL(V )).)

Remark . The group AutGE for the vector bundle E with G-structure has the same sig-nificance as the group of all self-diffeomorphisms of M for a smooth manifold M or thegroup of all linear isometries of an inner product vector space. I.e. AutGE is the ‘groupof natural symmetries of E’ and properties any objects one considers on the vector bundleare geometrically meaningful if they are preserved by this symmetry group.

One more remark on bundle automorphisms. A map hα giving the local expression overUα ⊆ B for a bundle automorphism may be interchangeably viewed as a transformationfrom one system of local trivializations to another. Any given local trivialization, say Φα

over Uα, is replaced by Φ′α. We have Φ′

α(e) = hα(π(e))Φα(e), e ∈ E. Respectively, thetransition functions are replaced according to ψ′βα = hβψβαh

−1α (point-wise group multi-

plication in the right-hand side). This is quite analogous to the setting of linear algebrawhere one may fix a basis and rotate the vector space, or fix a vector space and vary thechoice of basis—both operations being expressed as a non-singular matrix.

In Mathematical Physics (and now also in some areas of modern Differential Geometry)the group of G-bundle automorphisms is also known as the group of gauge transforma-tions4, sometimes denoted G.

4...and informally the ‘group of gauge transformations’ is often abbreviated as the ‘gauge group’ of E,although the ‘gauge group’ is really a different object! (It is the structure group G of vector bundle.) Alas,there is a danger of confusion.

28 alexei kovalev

2.2 Connections.

Sections of a vector bundle generalize vector-valued functions on open domains in Rn. Isthere a suitable version of derivative for sections, corresponding to the differential in multi-variate calculus? In order to propose such a derivative, it is necessary at least to understandwhich sections are to have zero derivative, corresponding to the constant functions on Rn.(Note that a section which is expressed as a constant in one local trivialization need notbe constant in another.)

Vertical and horizontal subspaces.

Consider a vector bundle π : E → B with typical fibre Rm and dimB = n. Let U ⊆ B bea coordinate neighbourhood in B and also a trivializing neighbourhood for E. Write xk,k = 1, . . . , n for the coordinates on U and aj, j = 1, . . . ,m for the standard coordinateson Rm. Then with the help of local trivialization the tangent space TpE for any point p,such that π(p) ∈ U , has a basis ∂

∂xk ,∂∂aj . The kernel of the differential (dπ)p : TpE → TbB,

b = π(p), is precisely the tangent space to the fibre Eb ⊂ E, spanned by ∂∂aj .

Definition. The vector space Ker (dπ)p is called the vertical subspace of TpE, denotedTvpE. A subspace Sp of TpE is called a horizontal subspace if Sp ∩ TvpE = 0 andSp ⊕ TvpE = TpE.

Thus any horizontal subspace at p is isomorphic to the quotient TpE/TvpE and hasthe dimension dim(TpE) − dim(TvpE) = dimB. Notice that, unlike the vertical tangentspace, a horizontal space can be chosen in many different ways (e.g. because there aremany choices of local trivialization near a given point in B).

It is convenient to specify a choice of a horizontal subspace at every point of E as thekernel of a system of differential 1-forms on E, using the following

Fact from linear algebra: if θ1, . . . , θm ∈ (Rn+m)∗ are linear functionals then onewill have dim(∩mi=1Ker θi) = n if and only if θ1, . . . , θm are linearly independentin (Rn+m)∗.

Now let θ1p, . . . , θ

mp be linearly independent ‘covectors’ in T ∗

pE, p ∈ π−1(U), and define

Sp := v ∈ TpE| θip(v) = 0, i = 1, . . . ,m.

We can write, using local coordinates on U ,

θip = f ikdxk + gijda

j, i = 1, . . . ,m, f ik, gij ∈ R. (2.7)

and any tangent vector in TpE as v = Bk( ∂∂xk )p + Ci( ∂

∂ai )p, Bk, Ci ∈ R. The θjp cannot all

vanish on a vertical vector, i.e. on a vector having Bk = 0 for all k. That is,

if gijCj = 0 for all i = 1, . . . ,m then Ci = 0 for all i = 1, . . . ,m.


Therefore the m × m matrix g = (gij) must be invertible. Denote the inverse matrix by

c = g−1, c = (cil). Replace θi by θi = cilθl = dai + eikdx

k, this does not change the space Sp.The above arrangement can be made for every p ∈ π−1(U), with f ik(p) and gij(p)

in (2.7) becoming functions of p. Call a map p 7→ Sp a field of horizontal subspaces ifthe functions f ik(p) and gij(p) are smooth. To summarize,

Proposition 2.8. Let S = Sp, p ∈ E, be an arbitrary smooth field of horizontal subspacesin TE. Let xk, aj be local coordinates on π−1(U) arising, as above, from some local trivial-ization of E over a coordinate neighbourhood U . Then Sp is expressed as Sp = ∩mj=1Ker θjp,where

θj = daj + ejk(x, a)dxk, (2.9)

for some smooth functions ejk(x, a). These ejk(x, a) are uniquely determined by a localtrivialization.

Definition. A field of horizontal subspaces Sp ⊂ TpE is called a connection on E if inevery local trivialization it can be written as Sp = Ker (θ1

p, . . . , θmp ) as in (2.9), such that

the functions eik(p) = eik(x, a) are linear in the fibre variables,

eik(x, a) = Γijk(x)aj. (2.10a)

and so

θip = dai + Γijk(x)ajdxk, (2.10b)

where Γijk : U ⊂ B → R are smooth functions called the coefficients of connection Spin a given local trivialization.

As we shall see below, the linearity condition in ai ensures that the horizontal sections(which are to become the analogues of constant vector-functions) form a linear subspace ofthe vector space of all sections of E. (A very reasonable thing to ask for.)

I will sometimes use an abbreviated notation

θip = dai + Aijaj, where Aij = Γijkdx

k,

So Proposition 2.8 identifies a connection with a system of matrices A = (Aij) of differential1-forms, assigned to trivializing neighbourhoods U ⊂ B.

The transformation law for connections.

Now consider another coordinate patch U ′ ⊂ X, U ′ = (xk′) and a local trivializationΦ′ : π−1(U ′) → U ′ × Rm with xk

′, ai

′the coordinates on U ′ × Rm.

Notation: throughout this subsection, the apostrophe ′ will refer to the local trivializationof E over U ′, whereas the same notation without ′ refers to similar objects in the localtrivialization of E over U . In particular, the transition (matrix-valued) functions from U

30 alexei kovalev

to U ′ are written as ψi′i and from U ′ to U as ψii′ , thus the matrix (ψii′) is inverse to (ψi

′i ).

Likewise (∂xk/∂xk′) denotes the inverse matrix to (∂xk

′/∂xk).

Recall that on U ′ ∩ U we have

xk′= xk

′(x), ai

′= ψi

′

i (x)ai. (2.11)

Then

dxk′=∂xk

′

∂xkdxk, dai

′= dψi

′

i ai + ψi

′

i dai

=∂ψi

′i

∂xkaidxk + ψi

′

i dai,

so

θi′= dai

′+ Γi

′

j′k′aj′dxk

′= dψi

′

i ai + ψi

′

i dai +

∂xk′

∂xkΓi

′

j′k′aj′dxk,

and

ψii′θi′ = dai + (ψii′ ·

∂ψi′j

∂xk+ ψii′ ·

∂xk′

∂xk· Γi′j′k′ · ψ

j′

j )ajdxk.

But then ψii′θi′ = θi and we find, by comparing with (2.10b), that

Γijk = Γi′

j′k′ψii′ψ

j′

j

∂xk′

∂xk+ ψii′

∂ψi′j

∂xk(2.12a)

and, using Aij = Γijkdxk,

Ai′

j′ = ψi′

i Aijψ

jj′ + ψi

′

i dψij′ , (2.12b)

Writing AΦ and AΦ′for the matrix-valued 1-forms expressing the connection A in the local

trivializations respectively Φ and Φ′ and abbreviating (2.11) to Φ′ = ψΦ for the transitionfunction ψ we obtain from the above that

AψΦ = ψAΦψ−1 + ψdψ−1 = ψAΦψ−1 − (dψ)ψ−1. (2.12c)

The above calculations prove.

Theorem 2.13. Any system of functions Γijk, i, j = 1, . . . ,m, k = 1, . . . , n attached to thelocal trivializations and satisfying the transformation law (2.12) defines on E a connectionA, whose coefficients are Γijk.

Remark . Suppose that we fix a local trivialization Φ and a connection A on E and regardψ as a bundle automorphism of E, ψ ∈ AutE. With his shift of view, the formula (2.12c)expresses the action of the group AutE on the space of connections on E. (Cf. the remarkon bundle automorphisms and linear algebra, page 25.)

Before considering the third view on connections we need a rigorous and systematicway to consider ‘vectors and matrices of differential forms’.


The endomorphism bundle EndE. Differential forms with values in vectorbundles.

Let (E,B, π) be a vector bundle with typical fibre V and transition functions ψβα. If Gis a linear map V → V , or endomorphism of V , G ∈ EndV , in a trivialization labelledby α, then the same endomorphism in trivialization β will be given by ψβαGψαβ (recallthat ψαβ = ψ−1

βα).This may be understood in the sense that the structure group of V acts linearly on

EndV . Exploiting, once again, the idea of Theorem 2.4 and the accompanying remarksone can construct from E a new vector bundle EndE, with the same structure group asfor E and with a typical fibre EndV . This is called the endomorphism bundle of avector bundle E and denoted EndE.

One can further extend the above construction and define over B the vector bundlewhose fibres are linear maps TbB → Eb, b ∈ B or, more generally, the antisymmetricmultilinear maps TbB × . . . × TbB → Eb on r-tuples of tangent vectors. (So the typicalfibre of the corresponding bundle is the tensor product Λr(Rn)∗ ⊗ V i.e. the space ofantisymmetric multilinear maps (Rn)r → V .) The sections of these bundles are calleddifferential 1-forms (respectively r-forms) with values in E and denoted Ωr

B(E). In anylocal trivialization, an element of Ωr

B(E) may be written as a vector whose entries aredifferential r-forms. (The ‘usual’ differential forms correspond in this picture to the casewhen V = R and E = B × R.)

In a similar manner, one introduces the differential r-forms ΩrB(EndE) with values in

the vector bundle EndE. These forms are given in a local trivialization as m×m matriceswhose entries are the usual differential r-forms. The operations of products of two matrices,or of a matrix and a vector, extend to Ωr

B(E) and ΩrB(EndE), in the obvious way, using

wedge product between the entries.Now from the examination of the transformation law (2.12) we find that although a

connection is expressed by a differential form in a local trivialization, a connection is notin general a well-defined differential form. The difference between two connectionshowever is a well-defined ‘matrix of 1-forms’, more precisely an element in Ω1

B(EndE).Thus the space of all connections on a given vector bundle E is naturally an affine

space. Recall that an affine space is a space of points and has a vector space assigned toit and the operation of ‘adding a vector to a point to obtain another point’ (with some‘obvious’ axioms imposed). The vector space assigned to the space of connectiond on E isΩ1B(EndE).

Covariant derivatives

Definition . A covariant derivative on a vector bundle E is a R-linear operator∇E : Γ(E) → Γ(T ∗B ⊗ E) satisfying a ‘Leibniz rule’

∇E(fs) = df ⊗ s+ f∇Es (2.14)

for any s ∈ Γ(E) and function f ∈ C∞(B).

32 alexei kovalev

Here I used Γ(·) to denote the space of sections of a vector bundle. Thus Γ(E) = Ω0B(E)

and Γ(T ∗B ⊗ E) = Ω1B(E).

Example. Consider a connection A and put ∇E = dA defined in a local trivialization by

dAs = ds+ As, s ∈ Γ(E).

More explicitly, one can write s = (s1, . . . , sm) with the help of a local trivialization, wheresj are smooth functions on the trivializing neighbourhood, and then

dA(s1, . . . , sm) =((∂s1

∂xk+ Γ1

jksj)dxk, . . . , (

∂sm

∂xk+ Γmjks

j)dxk).

The operator dA is well-defined as making a transition to another trivialization we haves = ψs′ and A = ψA′ψ−1 − (dψ)ψ−1, which yields the correct transformation law fordAs = ds+ As = d(ψs′) + (ψA′ψ−1 − (dψ)ψ−1)s′ = ψ(ds′ + A′s′) = ψ(dAs)

′.

Theorem 2.15. Any covariant derivative ∇E arises as dA from some connection A.

Proof (gist). Firstly, any covariant derivative ∇E is a local operation, which means thatis s1, s2 are two sections which are equal over an open neighbourhood U of b ∈ B then(∇Es1)|b = (∇Es2)|b. Indeed, let U0 be a smaller neighbourhood of b with the closureU0 ⊂ U and consider a cut-off function α ∈ C∞(B), so that 0 ≤ α ≤ 1, α|U0 = 1,α|B\U = 0. Then 0 = d(α(s1− s2)) = (s1− s2)⊗ dα+α∇E(s1− s2), whence s1(b) = s2(b).So, it suffices to consider ∇E in some local trivialization of E.

The proof now simply produces the coefficients Γijk of the desired A in an arbitrarylocal trivialization of E, over U say. Any local section of E defined over U may be written,with respect to the local trivialization, as a vector valued function U → Rm (m beingthe rank of E). Let ej, j = 1, . . .m denote the sections corresponding in this way to theconstant vector-valued functions on U equal to the j-th standard basis vector of Rm. Thenthe coefficients of the connection A are (uniquely) determined by the formula

Γijk =((∇Eej)(

∂

∂xk))i, (2.16)

where ∇E(ej) is a vector of differential 1-forms which takes the argument a vector field. Weused a local coordinate vector field ∂

∂xk(which is well-defined provided that a trivializing

neighbourhood U for E is also a coordinate neighbourhood for B) and obtained a localsection of E, expressed as a smooth map U → V . Then Γijk ∈ C∞(U) is the i-th componentof this map in the basis ej of V .

It follows, from the R-linearity and Leibniz rule for ∇E, that for an arbitrary localsection we must have

∇Es = ∇E(sjej) = (dsi + sjΓijkdxk)ei = dAs

where as usual A = (Aij) = (Γijkdxk), so we recover the dA defined above. It remains to ver-

ify that Γijk’s actually transform according to (2.12) in any change of local trivialization, sowe get a well-defined connection. The latter calculation is straightforward (and practivallyequivalent to verifying that dA is well-defined independent of local trivialization.)


The definition of covariant derivative further extends to differential forms with valuesin E by requiring the Leibniz rule, as follows

dA(σ ∧ ω) = (dAσ) ∧ ω + (−1)qσ ∧ (dω), σ ∈ ΩqB(E), ω ∈ Ωr(B),

with ∧ above extended in an obvious way to multiply vector-valued differential forms andusual differential forms. It is straightforward to verify, considering local bases of sectionsej and differential forms dxk, that in any local trivialization one has dAσ = dσ + A ∧ σ,where σ ∈ Ωq

B(E).

Remark (Parallel sections). A section s of E is called parallel, or covariant constant,if dAs = 0. In a local trivialization over coordinate neighbourhood U the section s isexpressed as s = (s1(x), . . . , sm(x)), x ∈ U and the graph of s is respectively Σ =(xk, sj(x)) ∈ U × Rm : x ∈ B, a submanifold of U × Rm. The tangent spaces to Σ are

spanned by ∂∂xk + ∂sj

∂xk∂∂aj , for k = 1, . . . , n. We find that the 1-forms θis(x) = dai +Γijkdx

ksj,i = 1, . . . ,m, vanish precisely on the tangent vectors to Σ.

This is just the horizontality condition for a tangent vector to E and we see thata section s is covariant constant if and only if any tangent vector to the graph of s ishorizontal. Another form of the same statement: s : B → E defines an embedding of Bin E as the graph of s and the tangent space to the submanifold s(B) at p ∈ E is thehorizontal subspace at p (relative to A) if and only if dAs = 0.

To sum up, a connection on a vector bundle E can be given in three equivalentways:

(1) as a (smooth) field of horizontal subspaces in TE depending linearly on the fibrecoordinates, as in (2.10a);or

(2) as a system of matrix-valued 1-forms Aij (a system of smooth functions Γijk) assigned toevery local trivialization of E and satisfying the transformation rule (2.12) on the overlaps;or

(3) as a covariant derivative ∇E on the sections of E and, more generally, on the differentialforms with values in E.

2.3 Curvature.

Let A be a connection on vector bundle E and consider the repeated covariant differentia-tion of an arbitrary section (or r-form) s ∈ Ωr

B(E) (assume r = 0 though). Calculation ina local trivialization gives

dAdAs = d(ds+As)+A∧(ds+As) = (dA)s−A∧(ds)+A∧(ds)+A∧(As) = (dA+A∧A)s.

Thus dAdA is a linear algebraic operator, i.e. unlike the differential operator but dA, thedAdA commutes with the multiplication by smooth functions.

dAdA(fs) = fdAdAs, for any f ∈ C∞(B). (2.17)

34 alexei kovalev

Notice that the formula (2.17) does not make explicit reference to any local trivialization.We find that (dAdAs) at any point b ∈ B is determined by the value s(b) at that point. Itfollows that the operator dAdA is a multiplication by an endomorphism-valued differential2-form. (This 2-form can be recovered explicitly in coordinates similarly to (2.16), usinga basis ei say of local sections and a basis of differential 1-forms dxk in local coordinateson B.)

Definition. The form

F (A) = dA+ A ∧ A ∈ Ω2(B; EndE).

is called the curvature form of a connection A.

Definition. A connection A is said to be flat is its curvature form vanishes F (A) = 0.

Example. Consider a trivial bundle B × Rm, so the space of sections is just the vector-functions C∞(B; Rm). Then exterior derivative applied to each component of a vector-function is a well-defined linear operator satisfying Leibniz rule (2.14). The correspondingconnection is called trivial, or product connection. It is clearly a flat connection.

The converse is only true with an additional topological condition that the base B issimply-connected; then any flat connection on E induces a (global) trivialization E ∼= B ×Rm (Examples 3, Q5) and will be a product connection with respect to this trivialisation.

Bianchi identity.

Covariant derivative on a vector bundle E with respect to a connection A can be extended,in a natural way, to any section of B of EndE by requiring the following formula to hold,

(dAB)s = dA(Bs)−B(dAs),

for every section s of E. Notice that this is just a suitable form of Leibniz rule.The definition further extends to differential forms with values EndE, by setting for

every µ ∈ Ωp(B; EndE) and σ ∈ Ωq(B;E),

(dAµ) ∧ σ = dA(µ ∧ σ)− (−1)pµ ∧ (dAσ).

(Write µ =∑

k Bkωk and σ =∑

j sjηj.) It follows that a suitable Leibniz rule also holdswhen µ1 ∈ Ωp(B; EndE), µ2 ∈ Ωq(B; EndE) to give dA(µ1∧µ2) = (dAµ1)∧µ2 +(−1)pµ1∧(dAµ2). In the special case of trivial vector bundle, E = B×R and with dA = d the exteriordifferentiation, the above formuli recover the familiar results for the usual differential forms.

In particular, any µ ∈ Ω2(B; EndE) in a local trivialization becomes a matrix of 2-formsand its covariant derivative is a matrix of 3-forms given by

dAµ = dµ+ A ∧ µ− µ ∧ A.

Now, for any section s of E, we can write dA(dAdA)s = (dAdA)dAs, i.e. dA(F (A)s) =F (A)dAs and comparing with the Leibniz rule above we obtain.

Proposition 2.18 (Bianchi identity). Every connection A satisfies dAF (A) = 0.


2.4 Orthogonal and unitary connections

Recall that an orthogonal structure on a (real) vector bundle E defined as a family Φα oflocal trivializations (covering the base) of this bundle so that all the transition functions ψβαbetween these take values in the orthogonal group O(m), m = rkE. These trivializationsΦα are then referred to as orthogonal trivializations. There is a similar concept of a unitarystructure and unitary trivializations of a complex vector bundle.

In this case, the standard Euclidean or Hermitian inner product on the typical fibreRm or Cm yields a well-defined inner product on the fibres of E. (Cf. Examples 3 Q1.)

Definition. We say that A is an orthogonal connection relative to an orthogonal structure,respectively a unitary connection relative to a unitary structure, on a vector bundle E if

d〈s1, s2〉 = 〈dAs1, s2〉+ 〈s1, dAs2〉

for any two sections s1, s2 of E, where 〈·, ·〉 denotes the inner product on the fibres of E.

Proposition 2.19. An orthogonal connection has skew-symmetric matrix of coefficientsin any orthogonal local trivialization. A unitary connection has skew-Hermitian matrix ofcoefficients in any unitary local trivialization.

Proof.

0 = 〈dA(si1ei), sj2ej〉+ 〈si1ei, dA(sj2ej)〉 − d〈si1ei, s

j2ej〉 = (Aij + Aji )s

i1sj2 for any si1, s

j2,

where ei is the standard basis of Rm or Cm.

Corollary 2.20. The curvature form F (A) of an orthogonal (resp. unitary) connection Ais skew-symmetric (resp. skew-Hermitian) in any orthogonal (unitary) trivialization.

2.5 Existence of connections.

Theorem 2.21. Every vector bundle E → B admits a connection.

Proof. It suffices to show that there exists a well-defined covariant derivative∇E on sectionsof E. We shall construct a example of ∇E using a partition of unity.

Let Wα be an open covering of B by trivializing neighbourhoods for E and Φα thecorresponding local trivializations. Then on each restriction E|Wα we may consider atrivial product connection d(α) defined using Φα. Of course, the expression d(α)s will onlymake sense over all of B if a section s ∈ Γ(E) is equal to zero away from Wα. Now considera partition of unity ρi subordinate to Wα. The expressions ρis, ρid(i)s make sense over allof B as we may extend by zero away from Wi. Now define

∇Es :=∞∑i=1

d(i)(ρis) =∞∑i=1

ρid(i)s, (2.22)

where for the second equality we used Leibniz rule for d(i) and the property∑∞

i=1 ρi = 1(so

∑∞i=1 dρi = 0). The ∇E defined by (2.22) is manifestly linear in s and Leibniz rule for

∇E holds because it does for each d(i).

Part III: Differential geometry (Michaelmas 2003, 2004)


3 Riemannian geometry

3.1 Riemannian metrics and the Levi–Civita connection

Let M be a smooth manifold.

Definition. A bilinear symmetric positive-definite form

gp : TpM × TpM → R

defined for every p ∈ M and smoothly depending on p is called a Riemannian metricon M .

Positive-definite means that gp(v, v) > 0 for every v 6= 0, v ∈ TpM . Smoothly dependingon p means that for every pair Xp,Yp of C∞-smooth vector fields on M the expressiongp(Xp, Yp) defines a C∞-smooth function of p ∈ M .

Alternatively, consider a coordinate neighbourhood on M containing p and let xi,i = 1, . . . , dim M be the local coordinates. Then any two tangent vectors u, v ∈ TpMmay be written as u = ui( ∂

∂xi )p, v = vi( ∂∂xi )p and gp(u, v) = gij(p)uivj, where the functions

gij(p) = g((

∂∂xi

)p,(

∂∂xj

)p) express the coefficients of the metric g in local coordinates. One

often uses the following notation for a metric in local coordinates

g = gijdxidxj.

The bilinear form (metric) g will be smooth if and only if the local coefficients gij = gij(x)are smooth functions of local coordinates xi on each coordinate neighbourhood.

Example 3.1. Recall (from Chapter 1) that any smooth regularly parameterized surface Sin R3,

r : (u, v) ∈ U ⊂ R2 → r(u, v) ∈ R3.

is a 2-dimensional manifold (more precisely, we assume here that S satisfies all the definingconditions of an embedded submanifold). The first fundamental form1 Edu2 + 2Fdudv +Gdv2 is a Riemannian metric on S.

The following formulae are proved in multivariate calculus.• A curve on S may be given as γ(t) = r(u(t), v(t)), a ≤ t ≤ b. The length of γ is then

computed as∫ b

a|γ(t)|dt =

∫ b

a

√Eu2 + 2Fuv + Gv2 dt.

• The area of S is∫∫

U

√EG− F 2 du dv.

1E = (ru, ru), F = (ru, rv), G = (rv, rv) using the Euclidean inner product

36


Theorem 3.2. Any smooth manifold M can be given a Riemannian metric.

Proof. Indeed, M may be embedded in Rm by Whitney theorem. Then the restriction(more precisely, a pull-back) of the Euclidean metric of Rm to M defines a Riemannianmetric on M . (Examples 3, Q1).

Remark . A metric, being a bilinear form on the tangent spaces, can be pulled back via asmooth map, f say, in just the same way as a differential form. But a pull-back f ∗g of ametric g will be a well-defined metric only if f has an injective differential.

Remark . As a Riemannian metric on M is an inner product on the vector bundle TM ,Theorem 3.2 is also a consequence of Q7 of Examples 2.

Definition. A connection on a manifold M is a connection on its tangent bundle TM .

Recall that a choice of local coordinates x on M determines a choice of local trivial-ization of TM (using the basis vector fields ∂

∂xi on coordinate patches). The transitionfunction ϕ for two trivializations of TM is given by the Jacobi matrices of the correspond-ing change of coordinates (ϕi

i′) = ( ∂xi

∂xi′ ).

Let Γijk be the coefficients (Christoffel symbols) of a connection on M in local coordi-

nates xi. For any other choice xi′ of local coordinates the transition law on the overlapbecomes (cf. Chapter 2, eqn. (2.12a))

Γijk = Γi′

j′k′∂xi

∂xi′

∂xj′

∂xj

∂xk′

∂xk+

∂xi

∂xi′

∂2xi′

∂xj∂xk(3.3)

One can see from the above formula that if Γijk are the coefficients of a connection on M

then Γikj also are the coefficients of some well-defined connection on M (in general, this

would be a different connection).The difference T i

jk = Γijk − Γi

kj is called the torsion of a connection (Γijk). The trans-

formation law for T ijk is T i

jk = T i′

j′k′∂xi

∂xi′∂xj′

∂xj∂xk′

∂xk , thus the torsion of a connection is a well-defined antisymmetric bilinear map sending a pair of vector fields X, Y to a vector fieldT (X, Y ) = T i

jkXjY k on M .

Definition. A connection on M is symmetric if its torsion vanishes, i.e. if Γijk = Γi

kj.

Notation: given a connection (covariant derivative) D : Ω0M(TM) → Ω1

M(TM) and asmooth vector field X on M , we write DX for the composition of D and contraction of1-forms (in Ω1

M(TM)) with X. Thus DX : Ω0M(TM) → Ω0

M(TM) is a linear differentialoperator acting on vector fields on M . In local coordinates, it is expressed as (DXY )i =Xj∂jY

i + ΓijkY

jXk.Here is a way to define a symmetric connection independent of the local coordinates.

Proposition 3.4. A connection D is symmetric if and only if DXY −DY X = [X, Y ].

The proof is an (easy) straightforward computation.

38 alexei kovalev

Theorem 3.5. On any Riemannian manifold (M, g) there exists a unique connection Dsuch that(1) d(g(X, Y ))(Z) = g(DZX, Y ) + g(X, DZY ) for any vector fields X, Y, Z on M ; and(2) the connection D is symmetric.

D is called the Levi–Civita connection of the metric g.

The condition (1) in the above theorem is sometimes written more neatly as

dg(X, Y ) = g(DX, Y ) + g(X, DY ).

Proof. Uniqueness. The conditions (1) and (2) determine the coefficients of Levi–Civita in

local coordinates as follows. A ‘coordinate vector field’ ∂∂xi with constant coefficients has

covariant derivative D ∂∂xi = Γp

ik∂

∂xp dxk. The condition (1) with X = ∂∂xi , Y = ∂

∂xj , Z = ∂∂xk

gives∂

∂xkgij = Γp

ikgpj + Γpjkgip. (3.6a)

Cycling i, j, k in the above formula, one can write two more relations

∂

∂xjgki = Γp

kjgpi + Γpijgkp, (3.6b)

∂

∂xigjk = Γp

jigpk + Γpkigjp. (3.6c)

Let (giq) denote the inverse matrix to (giq), so Γpjkgqpg

iq = Γijk. Adding the first two

equations of (3.6) and subtracting the third, dividing by 2, and multiplying both sides ofthe resulting equation by (giq), one obtains the formula

Γijk =

1

2giq

(∂gqj

∂xk+

∂gkq

∂xj− ∂gjk

∂xq

)(3.7)

(also taking account of the symmetry condition (2)). Thus if Levi–Civita connection existsthen its coefficients in local coordinates are expressed in terms of the metric by (3.7).

Existence. In view of the above calculations it suffices to make sure that the for-mula (3.7) indeed gives a well-defined connection on M . This can be done by verifying thatthe Γi

jk’s transform in the right way, i.e. as in (3.3), under a change of local coordinates.

The transformation law for gij is gi′j′ =∂xi

∂xi′gij

∂xj

∂xj′ , by the usual linear algebra. Differ-

entiating this latter formula and using the similar formula for the induced inner producton the dual spaces, i.e. on the cotangent spaces to M , we can verify that the coefficientsgiven by (3.7) indeed transform according to (3.3) and so the Levi–Civita connection ofthe metric g on M is well-defined.

3.2 Geodesics on a Riemannian manifold

Let E → M be a vector bundle endowed with a connection (Γijk). A parameterized smooth

curve on the base M may be written in local coordinates by (xi(t). A lift of this curve


to E is locally expressed as (xi(t), aj(t)) using local trivialization of the bundle E to definecoordinates aj along the fibres. A tangent vector (x(t), a(t)) ∈ T(xi(t),aj(t))E to a liftedcurve will be horizontal (recall from the chapter 2, eqn. (2.10b)) at every t precisely whena(t) satisfies a linear ODE

ai + Γijk(x)ajxk = 0, (3.8)

where i, j = 1, . . . , rank E, k = 1, . . . , dim B.Now if E = TM then there is a canonical lift of any smooth curve γ(t) on the base, as

γ(t) ∈ Tγ(t)M .

Definition. A curve γ(t) on a Riemannian manifold M is called a geodesic if γ(t) atevery t is horizontal with respect to the Levi–Civita connection.

Thus we are looking at a special case of (3.8) when a = x. The condition for a pathin M to be a geodesic may be written explicitly in local coordinates as

xi + Γijk(x)xjxk = 0, (3.9)

a non-linear second-order ordinary differential equation for a path x(t) = (xi(t)) (herei, j, k = 1, . . . , dim M). By the basic existence and uniqueness theorem from the theoryof ordinary differential equations, it follows that for any choice of the initial conditionsx(0) = p, x(0) = a there is a unique solution path x(t) defined for |t| < ε for somepositive ε. Thus for any p ∈ M and a ∈ TpM there is a uniquely determined (at leastfor any small |t|) geodesic with this initial data (i.e. ‘coming out of p in the direction a’).Denote this geodesic by γp(t, a) (or γ(t, a) if this is not likely to cause confusion).

Proposition 3.10. If γ(t) is a geodesic on (M, g) then |γ(t)|g = const.

Proof. We shall first make precise sense of the equation

Dγ γ = 0 (3.11)

and show that (3.11) is satisfied if and only if γ(t) is a geodesic. The problem with (3.11)at the moment is that γ is not a vector field defined on any open set in M , but only alonga curve γ. We shall define an extension, still denoted by γ, on a coordinate neighbourhoodU of γ(0) as follows. It may be assumed, without loss, that γ(0) = (xi(0)) has x1(0) 6= 0.We may further assume, taking a smaller U if necessary, that γ∩U , is a graph of a smoothfunction x1 7→ (x2(x1), . . . , xn(x1)). In particular, x1(t) 6= 0 for any small |t| and also anyhyperplane x1 = x1

0, such that |x10 − x1(γ(0))| is small, meets the curve γ ∩ U in exactly

one point. Denote by π the corresponding projection along hyperplanes x1 = const ontoγ ∩ U . Define, for every p ∈ U , γ(p) = γ(π(p)) and then γ is a smooth vector field on U ,such that (γ)p = γ(t) if γ(t) = p ∈ U .

Now let Γijk be the coefficients of the Levi–Civita in the coordinates on U . So DZY =

(Z l∂lYi+Γi

jkYjZk)∂i for any vector fields Z = Z l∂l, Y = Y i∂i on U . Let Y = Z = γ. Then

at any point p = γ(t) we have Z l∂lYi = xl ∂xi

∂xl= xi by the chain rule. It follows that the

40 alexei kovalev

equation (3.9) is equivalent to (3.11) if the latter if restricted to the points of the curve γ.It can also be seen, by inspection of the above construction, that Dγ γ is independent ofthe choice of extension of γ(t) to a vector field on U .

We have γd(γ, γ)g = (Dγ γ, γ)g + (γ, Dγ γ)g from the defining properties of the Levi–Civita. Hence γd|γ|2g = 0, by (3.11). But, by the construction, |γ|g is a function of

x1 only, so γd|γ|2g = x1(t)d

dx1|γ|2g. As x1(t) 6= 0 on U by earlier assumption, we must have

(d/dx1)|γ|2g = 0 and |γ|g = const as we had to prove.

Examples. 1. On Rn with the Euclidean metric∑

(dxi)2 we have Γiik = 0, so the Levi–

Civita is just the exterior derivative D = d. The geodesics xi = 0 are straight linesγp(t, a) = p + at parameterized with constant velocity.

2. Consider the sphere Sn with the round metric (i.e. the restriction of the Euclideanmetric to Sn ⊂ Rn+1). Then p ∈ Sn and a ∈ TpS

n may be regarded as the vectorsin Rn+1. Suppose a 6= 0, then the orthogonal reflection L in the 2-dimensional subspaceP = spanp, p+a is an isometry of Sn. Now L preserves the metric and p and a, the datawhich determines the geodesic γp(·, a). As γp(·, a) is moreover uniquely determined it mustbe contained in the fixed point set of L. But the fixed point set in a curve, the great circleP ∩ Sn. We find that great circles, parameterized with velocity of constant length—andonly these—are the geodesics on Sn.

Observe that for any geodesic γp(t, a) and any real constant λ the path γp(λt, a) is alsoa geodesic and γp(λt, a) = γp(t, λa).

By application of a general result in the theory of ordinary differential equations, ageodesic γp(t, a) must depend smoothly on its initial conditions p, a. Furthermore, thereexist ε1 > 0 and ε2 > 0 independent of a and such that if |a| < ε1 then γp(t, a) exists forall −2ε2 < t < 2ε2. It follows that γp(1, a) is defined whenever |a| < ε = ε1ε2.

Definition. The exponential map at a point p of a Riemannian manifold (M, g) is

expp : a ∈ N ⊆ TpM → γ(1; p, a) ∈ M.

Proposition 3.12. (d expp)0 = id(TpM)

Proof. We use the canonical identification a ∈ TpM → ddt

(ta)|t=0 to define (d expp)0 as alinear map on TpM (rather than on T0(TpM)).

Let |a| < ε, so γp(t, a) = γp(1, ta) is defined for 0 ≤ t ≤ 1. Then we have

(d expp)0a = ddt

expp(ta)|t=0

= ddt

γp(1, ta)|t=0

= ddt

γp(t, a)|t=0

= γ(0, a) = a.

Corollary 3.13. The exponential map expm defines a diffeomorphism from a neighbour-hood of zero in TmM to a neighbourhood of m in M .


Proof. Apply the Inverse Mapping Theorem (page 8 of these notes).

Corollary 3.13 means that the exponential map defines near every point p of a Rieman-nian manifold a system of local coordinates—called normal (or geodesic) coordinatesat p. It is not difficult to see that the geodesics γp(t, a) are given in these coordinates byrays emanating from the origin.

It also makes sense to speak of geodesic polar coordinates at p ∈ M defined by thepolar coordinates on TpM via a diffeomorphism

f : (r,v) ∈ ]0, ε[×Sn−1 → expp(r v) ∈ M. (3.14)

Here ]0, ε[×Sn−1 is regarded as a subset in TpM ∼= Rn via the inner product g(p). If0 < r < ε then the image Σr = f(r × Sn−1 ⊂ TpM) of the metric sphere of radius ris well-defined on M and is called a geodesic sphere about p. (So Σr is an embeddedsubmanifold of M .) The following remarkable result asserts that ‘the geodesic spheres areorthogonal to their radii’.

Gauss Lemma. The geodesic γ(t; p, a) is orthogonal to Σr. Thus the metric g in geodesicpolar coordinates has local expression g = dr2 + h(r, v), where for any 0 < r < ε, h(r, v) isthe metric induced by g on Σr (by restriction).

Proof. Let X be an arbitrary smooth vector field on the unit sphere Sn−1 ⊂ TpM . Usepolar coordinates to make sense of X as a vector field (independent of r) on the puncturedunit ball B \ 0 ⊂ TpM . Define a vector field X(r,v) = rX(v) on B \ 0. The mapexpp induces a vector field Y (f(r,v)) = (d expp)rvX(r,v) on the punctured geodesic ballB′ \ p = expp(B \ 0) in M .

We shall be done if we show that Y is everywhere orthogonal to the radial vectorfield ∂

∂r. Note that, by construction, any geodesic from p is given in normal coordinates by

γ(t; p, a) = at, so γ(t; p, a)/|a| = ∂∂r

. Here |a| means the norm in the inner product gp onthe vector space TpM . By application of Corollary 3.13, the family γ(t; p, a), where |a| = 1,defines a smooth vector field on B′\p. Recall from (3.11) that Dγ γ = 0 for any geodesic γ,where D denotes the Levi–Civita covariant derivative. Also d

dtg( ∂

∂r, ∂

∂r) = d

dtg(γ, γ) = 0 by

Proposition 3.10, so g( ∂∂r

, ∂∂r

) = 1. It remains to show that g(Y, γ) = 0.Using the diffeomorphism f in (3.14) to go to polar geodesic coordinates, we obtain

DγY −DY γ = (df)(D ∂

∂rX −DX

∂∂r

)= (df)

d

drX = (df)(X/r) = Y/r,

with the help of Proposition 3.4. Therefore, we find

d

drg(Y, γ) = g(DγY, γ) + g(Y,Dγ γ) = g(DγY, γ) = g(DY γ +

1

rY, γ) =

1

rg(Y, γ).

as 2g(Dγ, γ) = d g(γ, γ) = 0 by Proposition 3.11. Thus ddr

G = G/r, where G = g(Y, γ).Hence G is linear in r and d

drG independent of r. But limr→0

ddr

G = limr→0 g(X, ∂∂r

) = 0, as(d expp)0 is an isometry by Proposition 3.12, and so g(Y, γ) = 0 and the result follows.

42 alexei kovalev

3.3 Curvature of a Riemannian manifold

Let g be a metric on a manifold M . The (full) Riemann curvature R = R(g) of g is, bydefinition, the curvature of the Levi–Civita connection of g. Thus R ∈ Ω2

M(End(TM)),locally a matrix of differential 2-forms R = 1

2(Ri

j,kldxl ∧ dxk), i, j, k, l = 1 . . . n = dim M .The coefficients (Ri

j,kl) form the Riemann curvature tensor of (M, g). Given two vec-tor fields X, Y , one can form an endomorphism R(X, Y ) ∈ End(TM); its matrix inlocal coordinates is R(X, Y )i

j = Rij,klX

kY l (as usual X = Xk∂k, Y = Y l∂l). DenoteRkl = R(∂k, ∂l) ∈ End(TM).

Recall that in local coordinates a connection (covariant derivative) may be written asd + A, with A = Γi

jkdxk = Akdxk. We write Dk = D ∂

∂xk= ∂

∂xk + Ak. The definition of the

curvature form of a connection (Chapter 2, p. 31) yields an expression in local coordinates

Rij,kl =

(DlDk

∂∂xj −DkDl

∂∂xj

)i, or Rkl = −[Dk, Dl], (3.15)

considering the coefficient at dxl ∧ dxk. Now DX = XkDk and so we have −[DX , DY ] =−[XkDk, X

lDl] = −Xk(∂kYl)Dl − XkY lDkDl + Y k(∂kX

l)Dl + XkY lDlDk = XkY lRkl −[X, Y ]lDl. We have thus proved

Lemma 3.16. R(X, Y ) = D[X,Y ] − [DX , DY ].

One also can combine (3.15) with (3.7) and thus obtain an explicit local expression forRi

j,kl in terms of the coefficients of the metric g and their first and second derivatives.It is convenient to consider Rij,kl = giqR

qj,kl, which defines a map on 4-tuples of vector

fields (X, Y, Z, T ) 7→ g(R(X, Y )Z, T ).

Proposition 3.17.

(i) Rij,lk = −Rij,kl = Rji,kl;

(ii) Rij,kl + Ri

k,lj + Ril,jk = 0;

(iii) Rij,kl = Rkl,ij.

Proof. (i) The first equality is clear. For the second equality, one has, from the definitionof the Levi–Civita connection, ∂gkl

∂xi = g(Di∂

∂xk , ∂∂xl ) + g( ∂

∂xk , Di∂

∂xl ), and further

∂2gkl

∂xj∂xi= g(DjDi

∂

∂xk,

∂

∂xl) + g(Di

∂

∂xk, Dj

∂

∂xl) + g(Dj

∂

∂xk, Di

∂

∂xl) + g(

∂

∂xk, DjDi

∂

∂xl).

The right-hand side of the above expression is symmetric in i, j as ∂2gkl

∂xi∂xj = ∂2gkl

∂xj∂xi . Theanti-symmetric part of the right-hand side (which has to be zero) equals Rij,kl + Rji,kl.

(ii) Firstly, (Dk∂

∂xj )i = Γi

jk = (Dj∂

∂xk )i, by the symmetric property of the Levi-Civita.The claim now follows by straightforward computation using (3.15).

(iii) Multiplying (ii) by giq gives Rij,kl + Rik,lj + Ril,jk = 0. Similarly, Rjk,li + Rjl,ik +Rji,kl = 0, Rkl,ij + Rki,jl + Rkj,li = 0, and Rli,jk + Rlj,ki + Rlk,ij = 0. Adding up the fouridentities and making cancellations using (i) (the ‘octahedron trick’) gives the requiredresult.


Corollary 3.18. The Riemann curvature tensor (Rij,kl)p defines, at any point p ∈ M asymmetric bilinear form on Λ2TpM .

There are natural ways to extract “simpler” quantities (i.e. with less components) fromthe Riemann curvature tensor.

Definition. The Ricci curvature of a metric g at a point p ∈ M , Ricp = Ric(g)p, is thetrace of the endomorphism v → Rp(x, v)y of TpM depending on a pair of tangent vectorsx, y ∈ TpM .

Thus in local coordinates Ric(p) is expressed as a matrix Ric = (Ricij), Ricij =∑

q Rqi,jq.

That is, the Ricci curvature at p is a bilinear form on TpM . A consequence of Proposi-tion 3.17(iii) is that this bilinear form is symmetric, Ricij = Ricji.

Definition. The scalar curvature of a metric g at a point p ∈ M , s = scal(g)p is asmooth function on M obtained by taking the trace of the bilinear form Ricij with respectto the metric g.

If local coordinates are chosen so that gij(p) = δij, then the latter definition meansthat s(p) =

∑i Ricii(p) =

∑i,j Rij,ij(p). For a general gij, the formula may be written

as s =∑

i gij Ricij, where gij is the induced inner product on the cotangent space with

respect to the dual basis, algebraically (gij) is the inverse matrix of (gij)).

3.3.1 Some examples

(1) It makes sense to consider the condition

Ric = λg (3.19)

for some constant λ ∈ R, as both the metric and its Ricci curvature are symmetric bilinearforms on the tangent spaces to M . When the condition (3.19) is satisfied, the Riemannianmanifold (M, g) is called Einstein manifold. In particular, if (3.19) holds with λ = 0 thenM is said to be Ricci-flat.

(2) Recall that if Σ is a surface in R3 (smooth, regularly parameterized by (u, v) in anopen set in R2) then there is a metric induced on Σ, expressed as the first fundamentalform Edu2 + 2Fdudv + Gdv2. The second fundamental form Ldu2 + 2Mdudv + Ndv2 isdefined by taking the inner products L = (ruu,n), M = (ruv,n), N = (rvv,n) with theunit normal vector to Σ, n = ru × rv/|ru × rv| (the subscripts u and v denote respectivepartial derivatives). The quantity

K =LN −M2

EG− F 2

is called the gaussian curvature of Σ. A celebrated theorema egregium, proved by Gauss,asserts that K is determined by the coefficients of first fundamental form, i.e. by the metricon Σ (and so K is independent of the choice of an isometric embedding of Σ in R3).

44 alexei kovalev

Taking up a general view on Σ as a 2-dimensional Riemannian manifold, one can checkthat 2(EG − F 2)−1R12,12 = s, the scalar curvature of Σ. From the results of the nextsection, we shall see (among other things) that the scalar curvature of a surface Σ is twiceits gaussian curvature s = 2K.

3.4 Riemannian submanifolds

When a manifold Mn is an embedded submanifold of a Riemannian manifold, say V n+r,the Riemannian metric gV on V induces, by restriction, a Riemannian metric gM on M .What is the relation between the Levi–Civita connection D of gV and the Levi–Civitaconnection D of gM?

To see this relation, it is convenient to consider the vector bundle E = ι∗(TV ) over M ,where ι : M → V is the embedding map. (Informally, E is just the restriction of TV to Mif the latter is regarded as a subset of V .)

In the next proposition, we write xk for local coordinates on M , yγ for local coordinateson V , and α, β, γ = 1, . . . , n + r.

Proposition 3.20. Any connection ∇ on V induces in a canonical way a connection on Ewith the coefficients Γα

βk = ∂yγ

∂xk Γαβγ, where Γα

βγ are the coefficients of ∇ and y = y(x) is thelocal expression of the embedding ι.

We shall still denote by ∇ the connection on E defined by the above proposition. Forp ∈ E, consider the tangent space TpE as a subspace of TpV and then the correspondinghorizontal subspace of TpE is just the intersection Sp ∩ TpE, where Sp ⊂ TpV is thehorizontal subspace for the connection on V .

There is also an interpretation in terms of the covariant derivatives (needed for theproof of Gauss–Weingarten formulae below). Any local vector field X on M (respectivelylocal section σ of E) can be extended smoothly to a local vector field X (respectively σ)on V . Then (∇X σ)|M = ∇Xσ, where in the left-hand side we use the connection on E. Inparticular, the right-hand side is independent of the choices of extensions X and σ.

Thus the connection ∇ on E makes natural sense from all three points of view. Notethat we did not require any metric to define this induced connection.

Each fibre Ex of E contains TxM as a subspace. Using now the metric on M we obtaina direct sum decomposition

Ex = TxM ⊕ (TxM)⊥. (3.21)

The disjoint union of the orthogonal complements tx∈M(TxM)⊥ forms a vector bundle ofrank r over M called the normal bundle of M in V , denoted NM/V . Exercise: verifythat NM/V is indeed a well-defined vector bundle (recall Theorems 1.8 and 2.4).

For any two vector fields X, Y on M , we can decompose the covariant derivative(∇XY )x = (∇XY )x + (h(X, Y ))x, according to (3.21), where h(X, Y ) is some sectionof NM/V . It turns out that ∇ is a well-defined covariant derivative (connection) on M andh is a bilinear map TxM × TxM → (TxM)⊥ (depending smoothly on x). Furthermore, inthe case when ∇ = D is the Levi–Civita connection on V we obtain.


Theorem 3.22 (Gauss formula). For any vector fields X, Y on M ,

DXY = DXY + II(X, Y ),

where D is the Levi–Civita connection of the induced metric on M , and II is a symmetricbilinear map called the second fundamental form of M in V .

Theorem 3.23 (Weingarten formula). For any vector field X on M and section ξ of thenormal bundle NM/V ,

DXξ = −SξX +∇′Xξ,

where for any ξ, Sξ is a endomorphism of the vector bundle TM called the shape operatorand ∇′ is a connection on NM/V . Furthermore, the shape operator is symmetric with respectto the induced Riemannian metric M ,

gM(SξX, Y ) = gM(X, SξY ) = gV (II(X, Y ), ξ),

for any vector field Y on M .

By direct application of the above, we can compute the Riemann curvature R = (Rij,kl)of M in terms of the curvature of the ambient manifold and the second fundamental form.

Theorem 3.24 (Gauss).

R(X, Y, Z, T ) = R(X, Y, Z, T ) + gV (II(X, Z), II(Y, T ))− gV (II(X, T ), II(Y, Z)).

Corollary 3.25. The curvature of a submanifold M of a flat manifold is determined bythe second fundamental form of M .

When M is a surface in the Euclidean R3, this is equivalent to theorema egregiumdiscussed in the previous section.

3.5 Laplace–Beltrami operator

Throughout this section M is a connected oriented Riemannian manifold of dimension n.Let g denote a metric on M and let the orientation be given by a nowhere-zero n-form ε.

Starting from the vector fields ∂∂x1 , . . .

∂∂xn at a point x in a coordinate neighbourhood U ,

we can apply Gram–Schmidt process with x as a parameter. Thus we obtain a new systemof (smooth) vector fields e1, . . . , en which give an orthonormal basis of tangent vectors on aperhaps smaller neighbourhood of p (still denote this neighbourhood by U). Let ω1, . . . ωn

on U be the dual 1-forms to e1, . . . , en, in the sense that

ωj(ei) = δij at any point in U.

Then ωj give at every point p of U a basis of T ∗p M , the dual basic to ej.The metric on M induces, for every p = 0, . . . , n an inner product on the bundle ΛpT ∗M

by making ωi1(x) ∧ . . . ∧ ωip(x) : 1 ≤ i1 < . . . < ip ≤ n an orthonormal basis of ΛpT ∗xM .

46 alexei kovalev

If ω′j is another system of local 1-forms, on another coordinate neighbourhood U ′ say,and ω′j are orthonormal at every point in U ′ then

ω′1 ∧ . . . ∧ ω′n = det(Φ) ω1 ∧ . . . ∧ ωn on U ′ ∩ U,

for some orthogonal matrix Φ (depending on x ∈ U ′ ∩ U). Assuming, as we can onan oriented M , that all the coordinate neighbourhoods are chosen so that the Jacobiansdet(Φ) are positive on the overlaps, we find that ω1∧ . . .∧ωn is a well-defined nowhere-zeron-form ωg on all of M . We can further ensure that ωg = aε for some positive functiona ∈ C∞(M). Then ωg is called the volume form of M .

In (positively oriented) local coordinates, ωg =√

det gijdx1 ∧ . . . ∧ dxn.

Definition. The Hodge star on M is a linear operator on the differential forms

∗ : ΛpT ∗xM → Λn−pT ∗xM,

such that for any two p-forms α, β ∈ ΛpT ∗xM one has α ∧ ∗β = 〈α, β〉g ωg(x), where ωg isthe volume form on M .

It follows that if ωi is an orthonormal basis of a cotangent space T ∗xM then necessarily∗(ω1 ∧ . . . ∧ ωp) = ωp+1 ∧ . . . ∧ ωn. In particular, ∗1 = ωg and ∗ωg = 1. By permutationsof indices and by linearity, the Hodge star is then uniquely determined for any differentialform on M . Further, it follows that ∗∗ = (−1)p(n−p) on the p-forms.

Using the Hodge star we construct a differential operator

δ : Ωp(M) → Ωp−1(M)

putting δ = (−1)n(p+1)+1 ∗ d∗ if p 6= 0 and δ = 0 on Ω0(M) = C∞(M).

Definition. The Laplace–Beltrami operator, or Laplacian, on M is a linear differen-tial operator ∆ : Ωp(M) → Ωp(M) given by

∆ = δd + dδ.

Straightforward computation shows that when M is the Euclidean Rn the definition

gives ∆f = − ∂2f

(∂x1)2− . . . − ∂2f

(∂xn)2for any smooth function f . For a general metric

g = (gij), the local expression becomes ∆gf = − 1√det g

∂

∂xj

(√det g gij ∂f

∂xi

).

Proposition 3.26. The operator δ is the adjoint2 of d in the sense that∫M

〈dα, β〉g ωg =

∫M

〈α, δβ〉g ωg,

for every compactly supported α ∈ Ωp−1(M), β ∈ Ωp(M).

2It is more correct to say that δ is the ‘formal adjoint’ of d for reasons that have to do with the Analysis.


Using the inner product on the spaces ΛpT ∗p M , p ∈ M , we can define an inner product onΩp(M), called the L2 inner product, by putting 〈α, β〉L2 =

∫M〈α, β〉g ωg. The inner product

makes each Ωp(M) into a normed space, with L2-norm defined by ‖α‖ = (〈α, α〉L2)1/2. Inparticular, α = 0 if and only if ‖α‖ = 0.

Thus Proposition 3.26 says that 〈dα, β〉L2 = 〈α, δβ〉L2 and, consequently, 〈∆α, β〉L2 =〈α, ∆β〉L2 . It follows immediately that the Laplace–Beltrami operator is self-adjoint.

A differential form α ∈ Ωp(M) is called harmonic if ∆α = 0.

Corollary 3.27. Every harmonic differential form on a compact manifold is closed andco-closed: ∆α = 0 if and only if both dα = 0 and δα = 0.

Proof. Integration by parts, 0 = 〈δdα + dδα, α〉L2 = 〈δα, δα〉L2 + 〈dα, dα〉L2 .

It is also easily checked that ∗∆ = ∆∗ on any Ωp(M). Therefore the Hodge star of anyharmonic form is again harmonic.

Hodge Decomposition Theorem. Let M be a compact oriented Riemannian manifold.For every 0 ≤ p ≤ dim M , the space Hp of harmonic p-forms is finite-dimensional. Fur-thermore, there are L2-orthogonal direct sum decompositions

Ωp(M) = ∆Ωp(M)⊕Hp

= dδΩp(M)⊕ δdΩp(M)⊕Hp

= dΩp−1(M)⊕ δΩp+1(M)⊕Hp

(where we formally put Ω−1(M) = 0).Remark: the compactness condition on M cannot be removed.3

Short summary of the proof. We need to introduce the concept of a weak solution of

∆ω = α. (3.28)

A weak solution of (3.28) is by definition, a linear functional l : Ωp(M) → R which is(i) bounded, |l(β)| ≤ C‖β‖, for some C > 0 independent of β, and(ii) satisfies l(∆ϕ) = 〈α, ϕ〉L2 .

Any solution ω of (3.28) defines a weak solution by putting lω(β) = 〈ω, β〉L2 .The proof of Hodge Decomposition Theorem requires some results from Functional

Analysis.

Regularity Theorem. Any weak solution l of (3.28) is of the form l(β) = 〈ω, β〉L2, forsome ω ∈ Ωp(M) (and hence defines a solution of (3.28)).

Compactness Theorem. Assume that a sequence αn ∈ Ωp(M) satisfies ‖αn‖ < C and‖∆αn‖ < C, for some C independent of n. Then αn contains a Cauchy subsequence.

3The reason is that certain results in Analysis fail on non-compact sets, but this is another story.

48 alexei kovalev

We shall assume the above two theorems (and the Hahn–Banach theorem below)without proof.

Compactness Theorem implies at once that Hp must be finite-dimensional (for,otherwise, there would exist an infinite orthonormal sequence of forms). As Hp is finite-dimensional, we can write an L2-orthogonal decomposition Ωp(M) = Hp ⊕ (Hp)⊥.

It is easy to see that ∆Ωp(M) ⊆ (Hp)⊥ (use Proposition 3.26). For the reverse inclusion,suppose that α ∈ (Hp)⊥. We want to show that the equation (3.28) has a solution.Assuming the Regularity Theorem, we shall be done if we obtain a weak solution l :Ωp(M) → R of (3.28).

Define l first on a subspace ∆Ωp(M), by putting l(∆η) = 〈η, α〉L2 . It is not hard tocheck that l is well-defined. Further, (ii) is automatically satisfied (on this subspace); weclaim that (i) holds too. To verify the latter claim, we show that l is bounded below on∆Ωp(M) using, once again, the Compactness Theorem.

In order to extend l to all of Ωp(M), we appeal to

Hahn–Banach Theorem. Suppose that L is a normed vector space, and L0 a subspaceof L, and l : L0 → R a linear functional satisfying l(x0) < ‖x0‖, for all x0 ∈ L0. Then lextends to a linear functional on L with l(x) < ‖x‖ for all x ∈ L.

Thus we obtain a weak solution of (3.28) and deduce that Ωp(M) = ∆Ωp(M) ⊕ Hp

as desired. The two other versions of the L2-orthogonal decomposition of Ωp(M) followreadily by application of Proposition 3.26.

Corollary 3.29. Every de Rham cohomology class a ∈ Hr(M) of a compact oriented Rie-mannian manifold M is represented by a unique harmonic differential r-form α ∈ Ωr(M),[α] = a. Thus Hr ∼= Hr(M).

Proof. Uniqueness. If α1, α2 are harmonic p-forms and α1 − α2 = dβ then ‖dβ‖2 =〈dβ, α1 − α2〉L2 = 〈β, δ(α1 − α2)〉L2 = 0.

Existence. If α is such that dδα = 0 then ‖δα‖ = 0. Hence any closed p-form mustbe in dΩp−1(M)⊕Hp.

Corollary 3.29 is a surprising result: an analytical object (harmonic forms) turns out tobe equivalent to a topological object (de Rham cohomology) via some differential geometry.Here is a way to see ‘why such a result can be true’.

A de Rham cohomology class, a ∈ Hr(M) say, can be represented by many differentialforms; consider the (infinite-dimensional) affine space

Ba = ξ ∈ Ωr(M) | dξ = 0, [ξ] = a ∈ Hr(M)= ξ ∈ Ωr(M)|ξ = α + dβ, for some β ∈ Ωr−1(M).

When does a closed form α have the smallest L2-norm amongst all the closed forms in agiven de Rham cohomology class Ba?


Such a form must be a critical point of the function F (α + dβ) = ‖α + dβ)‖2 on Ba, sothe partial derivatives of F in any direction should vanish. That is, we must have

0 =d

dt

∣∣∣∣t=0

〈α + t dβ, α + t dβ〉L2 = 2〈α, dβ〉L2 .

Integrating by parts, we find that∫

M〈δα, β〉g = 0 must hold for every β ∈ Ωr−1(M). This

forces δα = 0, and so the extremal points of F are precisely the harmonic forms α.

Page references for this chapter

all to Gallot–Hulin–Lafontaine except ∗ to Warner

Riemannian metrics — pp.52–53,

Levi–Civita connection — pp.69–71,

curvature of a Riemannian manifold — pp.107–108,111–112,155–156

geodesics, Gauss Lemma — pp.80–85,89–90,

Riemannian submanifolds — pp.217–220,∗the Laplacian and the Hodge Decomposition Theorem — [W], pp. 140–141,220–226.

Differential Geometry III Lecture Notes - A. Kovalev

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