Sci. PROLONGATIONS OF THE TANGENT · 2019. 8. 1. · LOVEJOYS. DAS DepartmentofMathematics Kent State University, Tuscarawas Campus NewPhiladelphia, Ohio44663 (Received Febraury21,

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

1. KIM, J.B., Notes on l-manifold, Tensor (N.S.), Vol. 29 (1975), 299-302.

2. YANO, K. & ISHIHARA, S., On integrability of a structure y satisfying Is+ l 0, Quart.J. Math. Oxford, Vol. 25 (1964), 217-222.

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.

6. CALABI, E., Metric Reimann Surfaces, Annals of Math Studies, No. 30, Princeton UniversityPress, Princeton, 1953, 77-85.

7. BEJANCU, A. & YANO, K., CR-submanifolds of a complex spe form, J. Diff. Geom. 16(1981), 137-145.

204 L.S. DAS

8. DAS, L.S. & UPADHYAY, M.D., F-structure manifold, Kyung Pook Math. J. 18, Korea(1978), 272-283.

9. DAS, L.S., Complete lift of F-structure manifold, Kyung Pook Math. J. 20, Korea (1980),231-237.

10. DAS, L.S., On differentiable manifold with F(K, _(_)K+I) structure of rank ’r’, RevistaMathematica, Univ. Nac. Tucuman, Argentina, Rev. Ser. A27, No 1-2 (1978), 277-283.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Documents

Sci. PROLONGATIONS OF THE TANGENT · 2019. 8. 1. · LOVEJOYS. DAS DepartmentofMathematics Kent State University, Tuscarawas Campus NewPhiladelphia, Ohio44663 (Received Febraury21,