Hypersurfaces of the tangent bundle of a two-dimensional Riemannian manifold

Introduction. The present author [6] has shown that an orientable hypersurface in an almost complex manifold can be given an induced almost contact structure, and studied conditions for the induced structure to be normal. On the other hand, K.Yano and S.Kobayashi [8] and K.Yano and S.Ishihara [9] have introduced the notions of vertical, complete and horizontal lifts of tensor fields and connections to tangent bundles. Above all, if the base manifold is almost complex, then the complete or horizontal lift of the structure defines an almost complex structure in the tangent bundle. It is also known, cf. [5], that the tangent bundle of a Riemannian manifold is given an almost Kahlerian structure.1) Basing on these two kinds of results, we shall be able to induce various kinds of almost contact structures into tangent sphere bundles. In this paper, we shall consider a class of hypersurfaces with some property in tangent bundles ; the class contains the tangent sphere bundle of a manifold. In §§1 to 3, we shall introduce a tensor field for the study of hypersurfaces in almost complex manifolds and state some theorems on normality of induced almost contact structures in the language of the tensor field. In • ̃• ̃4 to 8, various kinds of almost contact structures will be induced to hypersurfaces in tangent bundles and the normality of the structures will be discussed. In • ̃8, we shall show that, given the almost Kahlerien structure in the tangent bundle of a Riemannian manifold, the induced almost contact structure of the tangent unit-sphere bundule is K-contact if and only if the base manifold is of positive constant curvature.

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On contact structures of tangent sphere bundles

Structures on differentiable manifolds have been very intensively studied in recent years both in differential topology as well as in differential geometry. Along with almost complex structures a systematic examination was begun of almost contact, affinor, symplectic, quaternion, and many other structures, a simple listing of which would take up considerable space. Not all of these structures are of equal value, but there is no doubt that the study of manifolds with various structures is of interest from both the global as well as the local point of view; moreover, for certain structures the possibility was outlined for applying them to various aspects of geometry and mechanics.

Hypersurfaces of the tangent bundle of a two-dimensional Riemannian manifold

References

On contact structures of tangent sphere bundles

Structures On Differentiable Manifolds

Almost contact metric hypersurfaces of the tangent bundle of euclidean space

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