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# Unit tangent bundle

About: Unit tangent bundle is a(n) research topic. Over the lifetime, 1056 publication(s) have been published within this topic receiving 15845 citation(s).

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01 Jan 1976

TL;DR: In this paper, the tangent sphere bundle is shown to be a contact manifold, and the contact condition is interpreted in terms of contact condition and k-contact and sasakian structures.

Abstract: Contact manifolds.- Almost contact manifolds.- Geometric interpretation of the contact condition.- K-contact and sasakian structures.- Sasakian space forms.- Non-existence of flat contact metric structures.- The tangent sphere bundle.

1,173 citations

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TL;DR: Tangent spaces of a sub-Riemannian manifold are themselves sub-riemannians as mentioned in this paper, and they come with an algebraic structure: nilpotent Lie groups with dilations.

Abstract: Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G.

706 citations

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TL;DR: In this paper, a Riemannian metric on the tangent sphere-bundles of the manifold T{M] was introduced, and the geodesic flow on it was considered.

Abstract: H.Poincare used the tangent sphere-bundles of ovaloids in three dimensional Euclidean space, i.e. the phase spaces of the ovaloids, to prove the existence of certain closed geodesies on the ovaloids. He introduced a Riemannian metric on the tangent sphere-bundles and considered the geodesic flow on it. As the metric of tangent bundles of Riemannian manifolds seems to be important, we would like to study differential geometry of tangent bundles of Riemannian manifolds by introducing on it natural Riemannian metrics. In this papar we shall do it by restricting ourselves only to the tangent bundles T{M).

480 citations

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TL;DR: In this paper, the Eckmann-Frölicher tensor of the tangent bündle of a manifold is computed, which implies that the manifold is integrable if and only if the linear connection has vanishing torsion and curvature.

Abstract: If M is a differentiable ra-dimensional manifold and V a linear connection for M, then the 2 rc-dimensional manifold TM, which is the total space of the tangent bündle of M, admits an almost complex structure /, naturally determined by V *). (I learned of this almost complex structure, which occurs e. g. in the theory of partial differential equations on Riemannian manifolds, frorn Professor W. Ambrose. I wish to thank him very much for the stimulating conversations which I have had with him on that topic.) We shall give here a computation of the Eckmann-Frölicher torsion tensor for this almost complex structure /, which implies the following result: / is integrable if and only if the linear connection has vanishing torsion and curvature). An appendix is devoted to some questions on the geometry of the tangent bündle TM which arise in connection with the construction of / and which can be answered easily by methods similar to those which we have used in order to compute the EckmannFrölicher torsion tensor of /. We list here only two of these results: If g is a Riemann metric for M and V its Levi-Civita connection, then TM admits a canonical hermitian metric hg with respect to the almost complex structure / on TM, which is determined (see above) by VWe prove, (confer Appendix (iii)) that hg is kählerian. If V is any linear connection for M, then the distribution of the \"horizontal subspaces\" on TM is invariant under the action of the multiplicative group R* of non vanishing real numbers on TM. We prove (confer Appendix (iv)) that if oppositely an n-dimensional distribution on TM is given, which is invariant under the action of the group JR* on TM and which contains no nonzero vertical\" vector, then this distribution

333 citations

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254 citations