# Some Notes on Geodesics of Vertical Rescaled Berger Deformation Metric in Tangent Bundle

TL;DR: In this paper , the authors studied the geodesics on the tangent bundle with respect to the vertical rescaled Berger deformation metric over an anti-paraK{a}hler manifold.

Abstract: In this paper, we study the geodesics on the tangent bundle $TM$ with respect to the vertical rescaled Berger deformation metric over an anti-paraK\"{a}hler manifold $(M, \varphi, g)$. In this case, we establish the necessary and sufficient conditions under which a curve be geodesic with respect to this. Finally, we also present certain examples of geodesic.

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TL;DR: In this paper , the authors studied the geodesics on the tangent bundle with respect to the vertical rescaled Berger deformation metric over an anti-paraK{a}hler manifold.

Abstract: In this paper, we study the geodesics on the tangent bundle $TM$ with respect to the vertical rescaled Berger deformation metric over an anti-paraK\"{a}hler manifold $(M, \varphi, g)$. In this case, we establish the necessary and sufficient conditions under which a curve be geodesic with respect to this. Finally, we also present certain examples of geodesic.

2 citations

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TL;DR: In this article , the authors investigated all forms of Riemannian curvature tensors of the unit tangent bundle equipped with a deformed Sasaki metric and presented the formulas of the Levi-Civita connection.

Abstract: Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with a deformed Sasaki metric. In this paper, firstly we investigate all forms of Riemannian curvature tensors of $TM$ (Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature). Secondly, we study the geometry of unit tangent bundle equipped with a deformed Sasaki metric, where we presented the formulas of the Levi-Civita connection and also all formulas of the Riemannian curvature tensors of this metric.

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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).

523 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

368 citations

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TL;DR: Some natural metrics on the tangent and on the sphere tangent bundle of Riemannian manifold were constructed and studied via the moving frame method in this article, and some natural metrics were constructed on the manifold manifold on the basis of these metrics.

Abstract: Some «natural» metrics on the tangent and on the sphere tangent bundle of Riemannian manifold are constructed and studied via the moving frame method.

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81 citations