Isospectral Orbifolds - An Introduction To Spectral ...mtm.ufsc.br/~martin/weilandt-ufcg.pdf · The Manifold Setting Constructing Isospectral Manifolds Isospectral Orbifolds Summary

The Manifold Setting Constructing Isospectral Manifolds Isospectral Orbifolds Summary & Outlook

Isospectral OrbifoldsAn Introduction To Spectral Geometry on Orbifolds

Martin [email protected]

Humboldt-Universität zu BerlinBerlin Mathematical School

2009-04-03 / UFCG


Outline

1 The Manifold SettingThe Notion of a ManifoldThe Laplace operator

2 Constructing Isospectral ManifoldsIsospectral ToriSunada’s Theorem

3 Isospectral OrbifoldsOrbifoldsExamples of Isospectral Orbifolds


Outline





Manifolds as Basic Objects of Differential Geometry

Definition

A manifold (of dimension n) is a metric space M with thefollowing additional structure: For every point p ∈ M there is asubset U ⊂ M with p ∈ U, V ⊂ R

n and a bijective mapφ : U → V ⊂ R

n, which is continuous in both directions(+compatibility conditions)


Manifolds as Basic Objects of Differential Geometry

Definition

A manifold (of dimension n) is a metric space M with thefollowing additional structure: For every point p ∈ M there is asubset U ⊂ M with p ∈ U, V ⊂ R

n and a bijective mapφ : U → V ⊂ R

n, which is continuous in both directions(+compatibility conditions)

Example (n = 2)Surfaces in R

3, e.g.S2 := {x ∈ R

3; x21 + x2

2 + x23 = 1} ⊂ R

3

(2-dimensional sphere)


More Examples of Manifolds

Revolution Surfaces (n = 2)

Open subsets of Rn: For instance

the open ball B := {x ∈ Rn; ‖x‖ < 1}

is an n-dimensional manifold.


One More Structure on a Manifold

In every point x ∈ M we have a vector space TxM ofso-called tangent vectors:


One More Structure on a Manifold

In every point x ∈ M we have a vector space TxM ofso-called tangent vectors:

Every vector v ∈ TxM has a certain length ‖v‖(e.g. the usual length in R

3 if M is just a surface in R3).


Why This Lengthy Definition?

On manifolds we have a distance function (= metric).



On manifolds we have a distance function (= metric).The additional structures allow us to define:

the notion of differentiable functions, denoted by C∞,the integral

∫

U f of a function f : M → R over a subset U ofM,



On manifolds we have a distance function (= metric).The additional structures allow us to define:

the notion of differentiable functions, denoted by C∞,the integral

∫

U f of a function f : M → R over a subset U ofM,

The volume of a compact manifold M is given by∫

M 1.


A Word of Warning

Not every natural geometric objects is a manifold: Forinstance a cone cannot satisfy our compatibility conditionsin the tip.


A Word of Warning

Not every natural geometric objects is a manifold: Forinstance a cone cannot satisfy our compatibility conditionsin the tip.

Since the cone is not “smooth” at the tip, we call this point“singular”.

However, the cone without the tip is a 2-dimensionalmanifold.


Outline





The Laplace operator on Rn

Let {ei}ni=1 denote the standard basis of R

n. On Rn we have the

following differential operators:

For a function f ∈ C∞(Rn) the gradient of f is the vectorfield

grad f :=∑

i

∂f∂x i ei .



Let {ei}ni=1 denote the standard basis of R

n. On Rn we have the

following differential operators:

For a function f ∈ C∞(Rn) the gradient of f is the vectorfield

grad f :=∑

i

∂f∂x i ei .

For a vector field X =∑

i X iei (with X i ∈ C∞(Rn)) thedivergence of X is the function

div X :=∑

i

∂X i

∂x i .



The Laplacian of a function f ∈ C∞(Rn) is the function

∆f := − div ◦grad f = −

(

∂2

∂x21

+ . . . +∂2

∂x2n

)

f .

Note that we can use the same definition for complex-valuedfunctions (see the second example below).



The Laplacian of a function f ∈ C∞(Rn) is the function

∆f := − div ◦grad f = −

(

∂2

∂x21

+ . . . +∂2

∂x2n

)

f .

Note that we can use the same definition for complex-valuedfunctions (see the second example below).

Example

(n = 2) For f (x) = x31 − 3x1x2

2 we have

∆f (x) = −(3 · 2x1 − 3 · 2x1), i.e. ∆f = 0.

(n = 1) For f (x) = eix we have

∆f (x) = −(i2eix) = eix , i.e. ∆f = f .


The Laplace Operator on a Riemanian manifold

Definition

Now let M be a manifold. The operators grad and divgeneralize to operators on M. Just like on R

n, the equation∆ = −div ◦ grad defines an endomorphism

∆ : C∞(M, R)−→C∞(M, R),

the Laplace Operator on M.



Definition



∆ : C∞(M, R)−→C∞(M, R),


Since ∆ is linear, we can consider the eigenspaces (assubspaces of C∞(M)) and eigenvalues of ∆.The preceding example gave eigenfunctions to theeigenvalues 0 and 1, respectively.



Definition



∆ : C∞(M, R)−→C∞(M, R),


Since ∆ is linear, we can consider the eigenspaces (assubspaces of C∞(M)) and eigenvalues of ∆.The preceding example gave eigenfunctions to theeigenvalues 0 and 1, respectively.

Note that C∞(M) is an infinite-dimensional vectorspace. We need to study functional analysis.


Properties of ∆

Theorem

Let M be a compact manifold. Then

∫

Mf1∆f2 =

∫

M‖grad f‖2 =

∫

Mf2∆f1.

This implies that the eigenvalues of ∆ are non-negative.

The eigenspace of ∆ are finite-dimensional.


Properties of ∆

Theorem

Let M be a compact manifold. Then

∫

Mf1∆f2 =

∫

M‖grad f‖2 =

∫

Mf2∆f1.

This implies that the eigenvalues of ∆ are non-negative.

The eigenspace of ∆ are finite-dimensional.

Definition

Two manifolds M1, M2 are called isospectral, if the eigenvaluesand the corresponding dimensions of the eigenspaces of thetwo Laplace operators ∆1, ∆2 coincide.


Classical Questions/Results

Question

“Can one hear the shape of a drum” (Kac, 1966)In other words: Is the geometry of a manifold determined by theeigenvalues of ∆?


Classical Questions/Results

Question

“Can one hear the shape of a drum” (Kac, 1966)In other words: Is the geometry of a manifold determined by theeigenvalues of ∆?

Theorem

Let M1, M2 be two isospectral manifolds. Then the followingcoincide for M1 und M2:

The dimension: dim(M1) = dim(M2)

The volume: vol(M1) = vol(M2)

Certain other (more complicated) geometric properties.


Main Goal

Task

In order to find geometric properties which are not determinedby the eigenvalues of ∆, we need to construct pairs ofisospectral manifolds.


Main Goal

Task

In order to find geometric properties which are not determinedby the eigenvalues of ∆, we need to construct pairs ofisospectral manifolds.

Note that isometric manifolds are not interesting in this contextbecause:

they are identical from the point of view of a geometer andtherefore

the Laplacians have the same eigenvalues, i.e., they aretrivially isospectral.


Outline





Quotient Manifolds

Let M be a manifold and let G be a discrete (e.g. finite) group ofisometries acting on M. Two points p1, p2 ∈ M are regarded asequivalent, if there is g ∈ G such that gp1 = p2.


Quotient Manifolds

Let M be a manifold and let G be a discrete (e.g. finite) group ofisometries acting on M. Two points p1, p2 ∈ M are regarded asequivalent, if there is g ∈ G such that gp1 = p2. Denote the setof equivalence classes by M/G. If

Gp := {g ∈ G; gp = p} = {Id}

for all p ∈ M, then M/G carries a natural manifold structuresuch that the differentiable functions on M/G are given by

C∞(M/G) = C∞(M)G := {f ∈ C∞(M, R); f ◦ g = f ∀g ∈ G}.


Quotient Manifolds

Let M be a manifold and let G be a discrete (e.g. finite) group ofisometries acting on M. Two points p1, p2 ∈ M are regarded asequivalent, if there is g ∈ G such that gp1 = p2. Denote the setof equivalence classes by M/G. If

Gp := {g ∈ G; gp = p} = {Id}

for all p ∈ M, then M/G carries a natural manifold structuresuch that the differentiable functions on M/G are given by

C∞(M/G) = C∞(M)G := {f ∈ C∞(M, R); f ◦ g = f ∀g ∈ G}.

Restricting ∆ on M to this subspace of C∞(M) gives theLaplace operator

∆ : C∞(M/G) → C∞(M/G)

on M/G.


Special Flat Tori

Let M = Rn and let G = Z

n act on M by translations. ThenM/G is an n-dimensional flat torus T n:

(Identify opposite sides by translation.)


General Flat Tori

More generally, we can consider a so-called lattice

Λ =

{

n∑

i=1

tivi ; ti ∈ Z

}

for {vi}ni=1 a basis of R

n.

Then the quotient Rn/Λ is also called a torus.


Eigenfunctions on Flat Tori

To determine the eigenfunctions of ∆ on Rn/Λ let v ∈ R

n andset fv (x) = e2πi〈v ,x〉 as a function on R

n. Then

∂

∂x j fv (x) = 2πivj fv (x) and(

∂

∂x j

)2

fv (x) = −4π2v2j fv (x).

Therefore ∆fv = 4π2‖v‖2fv , i.e. fv is an eigenfunction on Rn

associated with the eigenvalue 4π2‖v‖2.


Eigenfunctions on Flat Tori

To determine the eigenfunctions of ∆ on Rn/Λ let v ∈ R

n andset fv (x) = e2πi〈v ,x〉 as a function on R

n. Then

∂

∂x j fv (x) = 2πivj fv (x) and(

∂

∂x j

)2

fv (x) = −4π2v2j fv (x).

Therefore ∆fv = 4π2‖v‖2fv , i.e. fv is an eigenfunction on Rn

associated with the eigenvalue 4π2‖v‖2.Next set

Λ∗ := {v ∈ Rn; 〈v , λ〉 ∈ Z ∀λ ∈ Λ}.

and observe that fv ∈ C∞(Rn/Λ) if and only if v ∈ Λ∗. It can beshown that these are all eigenfunctions on R

n/Λ. Hence thespectrum on the torus R

n/Λ is given by the multiset

{4π2‖v‖2; v ∈ Λ∗}.


The First Example of Isospectral Manifolds

Theorem (Milnor 1964)

There are lattices Λ1, Λ2 in R16 such that the tori R

16/Λ1 andR

16/Λ2 are isospectral but not isometric.

Idea of Proof: Take certain Λ1, Λ2 from a paper by Ernst Witt(1941) and observe that his calculations imply the result above.(Use the Poisson Summation Formula.)


The First Example of Isospectral Manifolds

Theorem (Milnor 1964)

There are lattices Λ1, Λ2 in R16 such that the tori R

16/Λ1 andR

16/Λ2 are isospectral but not isometric.

Idea of Proof: Take certain Λ1, Λ2 from a paper by Ernst Witt(1941) and observe that his calculations imply the result above.(Use the Poisson Summation Formula.)

Remark

Later on it could be shown that there are isospectral tori ofdimension 4, but not of lower dimension.

There are only a few manifolds like the torus for which wecan explicitly calculate the eigenfunctions A more general way to construct isospectral manifoldsmight be useful.


Outline





Sunada’s Theorem

Theorem (Sunada 1985)

Let M be a compact Riemannian manifold and let G be a finitegroup of isometries acting on M such thatGp := {g ∈ G; gp = p} = {Id} ∀p ∈ M. Moreover, let Γ1,Γ2 besubgroups of G such that there is a bijection φ : Γ1 → Γ2

satisfying

∀γ ∈ Γ1 ∃gγ ∈ G : φ(γ) = gγγg−1γ

. (C)

Then the two manifolds M/Γ1, M/Γ2 are isospectral.


Sunada’s Theorem

Theorem (Sunada 1985)

Let M be a compact Riemannian manifold and let G be a finitegroup of isometries acting on M such thatGp := {g ∈ G; gp = p} = {Id} ∀p ∈ M. Moreover, let Γ1,Γ2 besubgroups of G such that there is a bijection φ : Γ1 → Γ2

satisfying

∀γ ∈ Γ1 ∃gγ ∈ G : φ(γ) = gγγg−1γ

. (C)

Then the two manifolds M/Γ1, M/Γ2 are isospectral.

Remark

If we can choose the same g ∈ G for every γ ∈ Γ1, then M/Γ1

and M/Γ2 are isometric.What makes the theorem above interesting is the existence ofexamples where we cannot choose the same g for every γ...


Example (Gassmann (1926))

Choose G = S6 the permutation group on {1, . . . , 6} and recallthat two permutations are conjugate if and only if they have thesame cycle structure. Hence

Γ1 := {Id, (12)(34), (13)(24), (14)(23)} and (1)

Γ2 := {Id, (12)(34), (12)(56), (34)(56)} (2)

satisfy Sunada’s conditions.Moreover it can be shown that we cannot choose the same gfor every element of Γ1.What remains to do is realize G as a subgroup of an isometrygroup.


Sunada’s Theorem

Proof.

Let V ⊂ C∞(M) denote the eigenspace of ∆ to the eigenvalueλ and for Γ ⊂ G set V Γ = {f ∈ V ; γ∗(f ) := f ◦ γ = f ;∀γ ∈ Γ}. Wehave to show that dim V Γ1 = dim V Γ2 .


Sunada’s Theorem

Proof.

Let V ⊂ C∞(M) denote the eigenspace of ∆ to the eigenvalueλ and for Γ ⊂ G set V Γ = {f ∈ V ; γ∗(f ) := f ◦ γ = f ;∀γ ∈ Γ}. Wehave to show that dim V Γ1 = dim V Γ2 .The homomorphism p : V ∋ f 7→ 1

|Γ|

∑

γ∈Γ γ∗(f ) ∈ V Γ is aprojection, i.e. p|V Γ = idV Γ . Therefore

dim V Γ = tr

1|Γ|

∑

γ∗∈Γ

γ∗

=1|Γ|

∑

γ∈Γ

tr γ∗.


Sunada’s Theorem

Proof.

Let V ⊂ C∞(M) denote the eigenspace of ∆ to the eigenvalueλ and for Γ ⊂ G set V Γ = {f ∈ V ; γ∗(f ) := f ◦ γ = f ;∀γ ∈ Γ}. Wehave to show that dim V Γ1 = dim V Γ2 .The homomorphism p : V ∋ f 7→ 1

|Γ|

∑

γ∈Γ γ∗(f ) ∈ V Γ is aprojection, i.e. p|V Γ = idV Γ . Therefore

dim V Γ = tr

1|Γ|

∑

γ∗∈Γ

γ∗

=1|Γ|

∑

γ∈Γ

tr γ∗.

1|Γ2|

∑

γ2∈Γ2

tr γ∗2 =

1|Γ1|

∑

γ∗∈Γ1

tr φ(γ)∗(C)=

1|Γ1|

∑

γ∗∈Γ1

tr((gγγg−1γ

)∗)

=1

|Γ1|

∑

γ∈Γ1

tr((g∗γγ∗(g−1

γ)∗) =

1|Γ1|

∑

γ∈Γ1

tr γ∗


Outline





A Short Repetition

Recall the definition of a quotient manifold: G a group ofisometries on a manifold M with Gp = {g ∈ G; gp = p} = {Id}.Then

M/G is a manifold withC∞(M/G) = {f ∈ C∞(M); f ◦ g = f ∀g ∈ G}

∆ on M induces the Laplacian ∆ : C∞(M/G) → C∞(M/G)on M/G.


A Short Repetition

Recall the definition of a quotient manifold: G a group ofisometries on a manifold M with Gp = {g ∈ G; gp = p} = {Id}.Then

M/G is a manifold withC∞(M/G) = {f ∈ C∞(M); f ◦ g = f ∀g ∈ G}

∆ on M induces the Laplacian ∆ : C∞(M/G) → C∞(M/G)on M/G.

Observation

The condition Gp = {Id} was not necessary for the definition of∆ on M/G.

So maybe omitting the condition Gp = {Id} might give aninteresting geometric object...


Good Orbifolds

Definition

Let M be a manifold and let G be a discrete (e.g. finite) group ofisometries on M. Then M/G will be called a good orbifold and

C∞(M/G) := {f ∈ C∞(M); f ◦ g = f ∀g ∈ G}


Good Orbifolds

Definition

Let M be a manifold and let G be a discrete (e.g. finite) group ofisometries on M. Then M/G will be called a good orbifold and

C∞(M/G) := {f ∈ C∞(M); f ◦ g = f ∀g ∈ G}

An Example:

Let D = {x ∈ R2; ‖x‖ < 1} denote the

unit disc in R2 and let R be a rotation

around the origin 0 by π/2. ThenG = {Id, R, R2, R3} ≃ Z4 is a group ofisometries and D/G is an orbifold.


∆ on Orbifolds

∆ : C∞(M/G) → C∞(M/G) is defined as in the setting ofquotient manifolds.

The eigenspaces are again finite-dimensional.

The definition of isospectrality is also analogous to themanifold case.


∆ on Orbifolds

∆ : C∞(M/G) → C∞(M/G) is defined as in the setting ofquotient manifolds.

The eigenspaces are again finite-dimensional.

The definition of isospectrality is also analogous to themanifold case.

However, orbifolds have one additional structure in comparisonto manifolds...


Isotropy

Definition

Let M/G be an orbifold and p ∈ M. The isotropy (or thestabilizer) of M/G in p is given by the group

Gp := {g ∈ G; gp = p} ⊂ G.

In our example of the quotient D/G of the unit disc we have

G0 = {Id, R, R2, R3} ≃ Z4

Gp = {Id} for p ∈ D \ {0}


Isotropy

Definition

Let M/G be an orbifold and p ∈ M. The isotropy (or thestabilizer) of M/G in p is given by the group

Gp := {g ∈ G; gp = p} ⊂ G.

In our example of the quotient D/G of the unit disc we have

G0 = {Id, R, R2, R3} ≃ Z4

Gp = {Id} for p ∈ D \ {0}

Question

Do the eigenvalues of ∆ determine the (isomomorphismclasses of) isotropy groups on an orbifold?


Outline





Observation (Bérard)

The proof of Sunada’s Theorem did not use Gp = {Id}, hence italso applies to orbifols.

This observation has been used to prove the followingtheorems:


Observation (Bérard)

The proof of Sunada’s Theorem did not use Gp = {Id}, hence italso applies to orbifols.

This observation has been used to prove the followingtheorems:

Theorem (Gordon, Webb, Wolpert (1992))

There are isospectral plane domains (“drums”), i.e. the answerto Kac’s famous question is: No, one cannot hear the shape ofa drum.


Another Application of Sunada

A little less historic but more relevant to our question is:

Theorem (Shams, Stanhope, Webb (2006))

There are isospectral 27-dimensional orbifolds where themaximal isotropy groups are not isomorphic.


Another Application of Sunada

A little less historic but more relevant to our question is:

Theorem (Shams, Stanhope, Webb (2006))

There are isospectral 27-dimensional orbifolds where themaximal isotropy groups are not isomorphic.

To find examples of smaller dimension, one can use thefollowing generalization of tori...


An Explicit Formula for Flat Orbifolds

Let G be a discrete subgroup of isometries on Rn. Write

Lb(x) = x + b for b, x ∈ Rn. Every element of G has the form

BLb with B an n × n-matrix satisfying BT B = In and b ∈ Rn. Set

Λ = ker{G ∋ BLb 7→ B ∈ O(n)} ⊂ Rn, F = im{G ∋ BLb 7→ B ∈ O(n)} ⊂

and Λ∗ = {v ∈ Rn; 〈λ, v〉 ∈ Z ∀λ ∈ Λ}.


An Explicit Formula for Flat Orbifolds

Let G be a discrete subgroup of isometries on Rn. Write

Lb(x) = x + b for b, x ∈ Rn. Every element of G has the form

BLb with B an n × n-matrix satisfying BT B = In and b ∈ Rn. Set

Λ = ker{G ∋ BLb 7→ B ∈ O(n)} ⊂ Rn, F = im{G ∋ BLb 7→ B ∈ O(n)} ⊂

and Λ∗ = {v ∈ Rn; 〈λ, v〉 ∈ Z ∀λ ∈ Λ}.

Theorem (Miatello, Rossetti 2001)

The dimension of the ∆-eigenspace to the eigenvalue 4π2µ onthe orbifold R

n/G is given by

E4π2µ(∆G) =1

#F

∑

B∈F

eµ,B(G),

where eµ,B(G) =∑

v∈Λ∗, ‖v‖2=µ

Bv=v

e2πi〈v ,b〉 (b ∈ Rn s.t. BLb ∈ G).


Two isospectral orbifolds R3/G1 and R

3/G2An Application of the preceding theorem with Λ = Z

3 and #F = 4

(Identify top with bottom and sides with the same symbols.)Observation: The isotropy Z2 appears in both orbifolds;Z4 only in R

3/G1 and Z2 × Z2 only in R3/G2.


Maximal Isotropy Order

Observation

The isotropy groups of maximal order in the example abovehad the same order 4.

This leads us to the following

Question

Do the eigenvalues of ∆ determine the maximal isotropy order(=: mio)?


Maximal Isotropy Order

Observation

The isotropy groups of maximal order in the example abovehad the same order 4.

This leads us to the following

Question

Do the eigenvalues of ∆ determine the maximal isotropy order(=: mio)?

Using the same methods as above, one can actually constructa pair of isospectral orbifolds R

3/Λ1, R3/Λ2 with mio1 = 2 and

mio2 = 4 (see the references on the last slide).


Summary & Outlook

The eigenvalues of ∆ do not determine the maximalisotropy order mio.


Summary & Outlook

The eigenvalues of ∆ do not determine the maximalisotropy order mio.

Open question (and motivation for this research): Can amanifold (mio = 1) be isospectral to an orbifold withnon-trivial isotropies (mio > 2)?

One possible approach: Study so-called bad orbifolds.

Appendix

Further Reading I

M. Weilandt.Isospectral Orbifolds with different Isotropy OrdersDiplom Thesis, 2007.(Available on www.math.hu-berlin.de/~weilandt)

D. Schüth, J.P. Rossetti, M. Weilandt.Isospectral orbifolds with different maximal isotropy ordersAnnals of Global Analysis and Geometry 34 (4), 2008

Isospectral Orbifolds - An Introduction To Spectral ...mtm.ufsc.br/~martin/weilandt-ufcg.pdf · The Manifold Setting Constructing Isospectral Manifolds Isospectral Orbifolds Summary

Documents