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Unit tangent bundle
About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.
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TL;DR: In this article, it was proved that the Sasaki metric of a tangent bundle is geodesic if and only if the curvature tensor of the tangent bundles is geodiscrete.
Abstract: It is proved that if the intrinsic zero-index of the Sasaki metric of a tangent bundleTM
n
isk, thenk is even andM
n
is the metric product of a Riemannian manifoldM
n−k/2 by a Euclidean spaceE
k/2, whileTM
n
is the metric product ofTM
n−k/2 byE
k
. An expression is obtained for the second fundamental forms of the imbeddingTF
l
⊂TM
n
in terms of the second fundamental forms of the imbeddingF
l
⊂M
n
and the curvature tensor ofM
n
. It is proved thatTF
l is totally geodesic inTM
n
if and only ifF
l
is totally geodesic inM
n
.
TL;DR: In this paper, the authors considered the case of the tangent bundle over real, complex, and quaternionic space forms and gave a unified proof of the following property: all geodesic curvatures of the projected curve are zero beginning with k3, k6, and k10.
Abstract: It is well known that if Г is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve λ=π 1463-01 Г are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with k3, k6, and k10 for the real, complex, and quaternionic space forms, respectively.
TL;DR: In this article, the parall elism, incompressibility and closeness conditions of the complete lift of vector fields are investigated with respect to Kaluza-Klein metric on tangent bundle.
Abstract: In this paper, differential equations of geodesics; parall elism, incompressibility and closeness conditions of the h orizontal and complete lift of the vector fields are investigated with r espect to Kaluza-Klein metric on tangent bundle.
TL;DR: In this article, it was shown that on a 4-manifold M endowed with a spin C -structure induced by an almost-complex structure, a self-dual (positive) spinor field Φ E Γ(W + ) is the same as a bundle morphism Φ: T M → T M acting on the fiber by selfdual conformal transformations, such that the Clifford multiplication is just the evaluation of Φ on tangent vectors, and that the squaring map a: W + → Λ + acts by pulling-
Abstract: We show that, on a 4-manifold M endowed with a spin C -structure induced by an almost-complex structure, a self-dual (positive) spinor field Φ E Γ(W + ) is the same as a bundle morphism Φ: T M → T M acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of Φ on tangent vectors, and that the squaring map a: W + → Λ + acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kahler and symplectic structures.