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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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30 Oct 2013
TL;DR: In this paper, the authors considered a deformation of Sasaki metric on the tangent bundle of a Riemannian manifold and derived conditions for the deformation to be recurrent or pseudo symmetric.
Abstract: In the present paper, we consider a deformation (in the horizontal bundle) of Sasaki metric on the tangent bundle TM over an n dimensional Riemannian manifold (M;g): We rstly study some properties of deformed Sasaki metric which is pure with respect to some paracomplex structures on TM: Finally conditions for deformed Sasaki metric to be recurrent or pseudo symmetric are given.
12 citations
TL;DR: In this paper, it was shown that the curvature tensor field of the Levi-Civita connection on (TM°, G) is completely determined by the Vranceanu connection and some adapted tensor fields on TM°.
Abstract: Let Fm = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on TM°. We show that the curvature tensor field of the Levi-Civita connection on (TM°, G) is completely determined by the curvature tensor field of Vranceanu connection and some adapted tensor fields on TM°. Then we prove that (TM°, G) is locally symmetric if and only if Fm is locally Euclidean. Also, we show that the flag curvature of the Finsler manifold Fm is determined by some sectional curvatures of the Riemannian manifold (TM°, G). Finally, for any c ≠ 0 we introduce the c-indicatrix bundle IM (c) and obtain new and simple characterizations of Fm of constant flag curvature c by means of geometric objects on both IM (c) and (TM°, G).
12 citations
TL;DR: In this article, the authors define connections on a parabolic principal bundle and apply them to a set of applications in computer vision and artificial intelligence applications, where connections on the principal bundle are defined.
Abstract: The aim here is to define connections on a parabolic principal bundle. Some applications are given.
12 citations
12 citations
TL;DR: In this article, it was shown that the immersion problem for manifolds is not a cross section problem for the stable normal bundle, but a cross-section problem for a stable tangent bundle.
Abstract: 1. M. Hirsch [3] has shown that the immersion problem for manifolds is just a cross section problem for the stable normal bundle. Our object here is to find conditions under which sections of the tangent bundle will imply sections in the normal bundle (and conversely). First we need some notation. Given an integer /, let j(t) be the maximum integer such that the 2*-f old Whitney sum of the Hopf bundle over RP^~ is trivial. If £ is a stable bundle, let gd(£) denote the geometric dimension of £.
12 citations