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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, the authors introduce the concepts of time dependent connections and time dependent semisprays on a manifold and their induced vector bundle structures on the second order time dependent tangent bundle.
Abstract: The aim of this paper is to geometrize time dependent Lagrangian mechanics in a way that the framework of second order tangent bundles plays an essential role. To this end, we first introduce the concepts of time dependent connections and time dependent semisprays on a manifold $M$ and their induced vector bundle structures on the second order time dependent tangent bundle $\R\times T^2M$. Then we turn our attention to regular time Lagrangians and their interaction with $\R\times T^2M$ in different situations such as mechanical systems with potential fields, external forces and holonomic constraints. Finally we propose an examples to support our theory.
2 citations
TL;DR: In this article, it was shown that the holomorphic principal G-bundle over T given by f admits a flat holomorphic connection, where G is the group of all holomorphic automorphisms of a fiber.
Abstract: Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.
2 citations
01 Jan 2008
TL;DR: In this article, the authors studied the properties of the tangent bundles with metrics of general natural lifted type and found the conditions under which the Riemannian manifold with a given tangent bundle has constant sectional curvature.
Abstract: We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle of $M$ and $G$ is the general natural lifted metric of $g$, has constant sectional curvature.
2 citations
01 Jan 2013
TL;DR: The main aim of as discussed by the authors is to investigate curvature properties and geodesics of the g-natural metric on the cotangent bundle of Riemannian manifold.
Abstract: The main aim of this paper is to investigate curvature properties and geodesics of the g-natural metric on the cotangent bundle of Riemannian manifold.
2 citations
TL;DR: In this article, the authors studied the geometric properties of the base manifold for the tangent sphere bundle of radius r satisfying the η-Einstein condition with the standard contact metric structure.
Abstract: We study the geometric properties of the base manifold for the tangent sphere bundle of radius r satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the tangent sphere bundle of the n(≥3)-dimensional locally symmetric space, equipped with the standard contact metric structure, is an η-Einstein manifold if and only if the base manifold is a space of constant sectional curvature or .
2 citations