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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 paper, the authors studied the geometric properties of the base manifold for the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, if and only if base manifold is the space of constant sectional curvature 1 or 2.
Abstract: We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $\eta$-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is $\eta$-Einstein manifold if and only if base manifold is the space of constant sectional curvature 1 or 2.
3 citations
01 Jan 2014
TL;DR: In this article, a generalized almost tangent geometry on the big tangent bundle T big M associated to an almost-tangent structure on M is studied with a special view towards integrability, and the compatibility with a class of generalized Riemannian metrics is discussed.
Abstract: A generalized almost tangent structure on the big tangent bundle T big M associated to an almost tangent structure on M is con- sidered and several features of it are studied with a special view towards integrability. Deformation under a fl- or a B-fleld transformation and the compatibility with a class of generalized Riemannian metrics are discussed. Also, a notion of tangentomorphism is introduced as a difieomorphism f preserving the (generalized) almost tangent geometry and some remarka- ble subspaces are proved to be invariant with respect to the lift of f.
3 citations
3 citations
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TL;DR: In this paper, a family of Kahler metrics on the tangent disk bundle of any real space-form or any of its quotients by discrete groups of isometries is presented.
Abstract: We nd a family of Kahler metrics on the tangent disk bundle of any real space-form or any of its quotients by discrete groups of isometries. Both zero-section and bres embed as real Lagrangians and totally geodesic submanifolds. The metric is complete in the non-negative curvature case and non-complete in the negative curvature case.
3 citations
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TL;DR: In this paper, an adapted convected frame field is introduced as a means of locally representing the transversal geometry of a flow in terms of the information contained in the flow and its derivatives.
Abstract: The analogy between 1+3 splittings of the spacetime tangent bundle and the splitting of the tangent bundle to the bundle of linear frames into vertical and horizontal sub-bundles is described from the unifying standpoint of the geometry of foliations. The physical nature of the line field on spacetime that plays the role of vertical sub-bundle is discussed in some detail. The notion that the complementary spatial bundle is most fundamentally a representation of the normal bundle to the foliation of spacetime by the integral curves of the line field is proposed, such that the geometry of space becomes the transverse geometry of the flow and the integration of the spatial bundle into proper time simultaneity hypersurfaces becomes a secondary issue to the geometry of space as the geometry of the leaf space of the foliation. The concept of an adapted convected frame field is introduced as a means of locally representing the transversal geometry of a flow in terms of the information that is contained in the flow and its derivatives, and some discussion is given to the role of the Bott connection.
3 citations