Internat. J. Math. & Math. Sci. VOL. 16 NO. (1993) 201-204 PROLONGATIONS OF F-STRUCTURE TO. THE TANGENT BUNDLE OF ORDER 2 LOVEJOY S. DAS Department of Mathematics Kent State University, Tuscarawas Campus New Philadelphia, Ohio 44663 (Received Febraury 21, 1991 and in revised form March 29, 1991) 201 ABSTRACT. A study of prolongations of F-structure to the tangent bundle of order 2 has been presented. KEY WORDS AND PHRASES. Prolongations, tangent bundle, integrable, lift, F-structure. 1991 AMS SUBJECT CLASSIFICATION CODE. 53C15 1. INTRODUCTION. Let F be a nonzero tensor field of type (1,1) and of class c on an n-dimensional manifold V such that [1] F K +(-)K+IF--O and F TM +(-)W+IF #0 for <W <K (1.1) where K is a fixed positive integer greater than 2. Such a structure on V, is called an F-structure of rank ’r’ and degree K. If the rank of F is constant and r r(F), then v, is called an F-structure manifold of degree K( _> 3). The case when K is odd has been considered in this paper. Let the operators on v, be defined as follows [1]: )KF and m I + )K + F K (1.2) where I denotes the identity operator on V,. From the operators defined by (1.2) we have + m I and 12 l; and m m (1.3) For F satisfying (1.1), there exist complementary distributions L and M corresponding to the projection operators and m respectively. If rank (F) constant on V,, then dim L r and dim M (n- r). We have the following results [l Fi IF F and Fm= mF 0 (1.4a) FK-l -I and FC-m 0 (1.4b) 2. PROLONGATIONS OF F-STRUCTURE IN THE TANGENT BUNDLE OF ORDER 2. Let V, be an n-dimensional differentiable manifold of class c and T,(V,,)=eU v T,(V,,) is the tangent bundle over the manifold v,. t, Let us denote T;(V,), the set of all tensor fields of class c and of the type (r,s) in V, and T(V,) be the tangent bundle over V,.
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Internat. J. Math. & Math. Sci.VOL. 16 NO. (1993) 201-204
PROLONGATIONS OF F-STRUCTURE TO. THE TANGENTBUNDLE OF ORDER 2
LOVEJOY S. DAS
Department of MathematicsKent State University, Tuscarawas Campus
New Philadelphia, Ohio 44663
(Received Febraury 21, 1991 and in revised form March 29, 1991)
201
ABSTRACT. A study of prolongations of F-structure to the tangent bundle of order 2 has beenpresented.
KEY WORDS AND PHRASES. Prolongations, tangent bundle, integrable, lift, F-structure.
1991 AMS SUBJECT CLASSIFICATION CODE. 53C15
1. INTRODUCTION.Let F be a nonzero tensor field of type (1,1) and of class c on an n-dimensional manifold V
such that [1]
FK +(-)K+IF--O and FTM +(-)W+IF #0 for <W <K (1.1)
where K is a fixed positive integer greater than 2. Such a structure on V, is called an F-structureof rank ’r’ and degree K. If the rank of F is constant and r r(F), then v, is called an F-structure
manifold of degree K( _> 3). The case when K is odd has been considered in this paper.Let the operators on v, be defined as follows [1]:
)KF and m I + )K + FK (1.2)
where I denotes the identity operator on V,.From the operators defined by (1.2) we have
+ m I and 12 l; and m m (1.3)
For F satisfying (1.1), there exist complementary distributions L and M corresponding to theprojection operators and m respectively.
If rank (F) constant on V,, then dim L r and dim M (n- r). We have the following results
[lFi IF F and Fm= mF 0 (1.4a)
FK-l -I and FC-m 0 (1.4b)
2. PROLONGATIONS OF F-STRUCTURE IN THE TANGENT BUNDLE OF ORDER 2.
Let V, be an n-dimensional differentiable manifold of class c and T,(V,,)=eUv T,(V,,) is the
tangent bundle over the manifold v,. t,
Let us denote T;(V,), the set of all tensor fields of class c and of the type (r,s) in V, and
T(V,) be the tangent bundle over V,.
202 L.S. DAS
Let us introduce an equivalence relation in the set of all differentiable mappings F:RV,where R is the real line. Let r >_ be a fixed integer. If two mappings F: R-.V, and G: R--V, satisfy
the conditions
dFh(O) dGa(O) dr’(o)Fh(O) Gh(O), dt dt dt
the mapping F and G being represented respectively by Xh= Fh(t) and Xh =Gh(t), (t E R) with
respect to local coordinates x in a coordinate neighborhood {U,X} containing the point
P F(0)=G(0), then we say that the mapping F is equivalent to G. Eah equivalence class
determined by the equivalence relation is called an r-jet of V. and denoted by J(F). The set of
all r-jets of V. is called the tangent bundle of order r and denoted by T.(Vn). The tangent bundle
T2(V.) of order 2 has the natural bundle structure over Vn, its bundle projection
being defined Iy 2(Jp2(F))= P. If we introduce a mapping such that P F(0), then T(V.) has a
bundle structure over T(Vn) with projectionLet us denote T(Vn) the second order tangent bundle over v and let Fu be the second lift of
F in T(Vn). The second lift Fu which belong to T(T(Vn) has component of the form [3]
F 0 0
FII:
".F + (/)’’,.F ’.F F
with respect to the induced coordinates in T2(Vn), F being local components of F in V..Now we obtain the following results on the second liit of F satisfying (1.1).For any F,G T(Vn), the following holds [3]:
(GIIFII)XII GII(FXII),
:GII(Fx)II
:(G(FX))II
(GF)fIXII for every X T(Vn)
therefore we have
(2.1)
(2.2)
GUFII (GF)I
If P(s) denote a polynomial of variable s, then we have
(P(F))II p(FII), where F T](Vn) (2.3)
We have the following theorem:
THEOREM 2.1. The second lift Fu defines a F-structure in T(V.) iff F defines a F-structure
in V,.PROOF. Let F satisfy (1.1) then F defines F-structure in V. satisfying
FK
which in view of equation (2.3) yields
PROLONGATIONS OF F-STRUCTURE TO THE TANGENT BUNDLE 203
(FH)K + )K + 1FH 0.
Therefore FH defines a F-structure in T2(Vn). The converse can be proved in a similar manner.
THEOREM 2.2. The second lift F is integrable in T2(V,,), iff F is integrable in Vn.PROOF. Let us denote NH and N, the Nijenhuis tensors of FH and F respectively. Then we
have [2]
N11(X,Y) (N(X,Y))11 (2.5)
We know that F-structure is integrable in V,, iff
which in view of (2.5) is equivalent to
N(X,V)=O,
NII(X,Y =0. (2.6)
Thus FH is integrable, iff F is integrable in V,,.THEOREM 2.3. The second lift FH of F is partially integrable in T2(V,,), iff F is integrable in
V
PROOF. We know that for F to be partially integrable in V,,the following holds [2]:
N(X, tV) oand
N(mX, mY) O,
which, in view of equation (2.5), takes the form
NII(IIIXII, III, yII) 0and (2.7)
NII(mlIXII, mlIyII) O.
where ill, mII axe operators in T(V,,) which define the distribution LH and MII respectively. Thus
equation (2.7) gives the condition for F to be partially integrable.The converse follows in a similar manner.
REFERENCES
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3. YANO, K. & ISHIHARA, S., Tangent and Cotangent Bundles, Marcel Dekkar, Inc., NewYork, 1973.
4. DOMBROWSKI, P., On the geometry of the tangent bundle, J. Reine Angewandte Math. 210(1980), 73-80.
5. HELGASON, S., Differential Geometry, Lie Groups and Symmetric Spac.es, Academic Press,New York, 1970.
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