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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 following theorem is proved: if M is an n-dimensional (n>2) submanifold of a Riemannian manifold N, and through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p, then M is totally geodesic in N.
Abstract: The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N or an extrinsic sphere of N.
2 citations
TL;DR: In this article, it was shown that there is a natural homomorphism between the Lie algebras of fiber preserving Killing vector fields on these bundles for any orientable Riemannian manifold of dimension two.
Abstract: Given an orientable Riemannian manifold, we consider the bundle of oriented orthonormal frames and the tangent sphere bundle over it, which admit natural Riemannian metrics defined by the Riemannian connection. We show that there is a natural homomorphism between the Lie algebras of fiber preserving Killing vector fields on these bundles. In particular, for any orientable Riemannian manifold of dimension two, we show that the homomorphism is extended to an isomorphism between these Lie algebras.
2 citations
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TL;DR: In this article, a noncommutative geometry with generalised tangent bundle from Fell bundle $C^*$-categories was constructed, by replacing pair groupoid objects (points) with objects in $E$.
Abstract: We construct a noncommutative geometry with generalised `tangent bundle' from Fell bundle $C^*$-categories ($E$) beginning by replacing pair groupoid objects (points) with objects in $E$. This provides a categorification of a certain class of real spectral triples where the Dirac operator is constructed from morphisms in a category. Applications for physics include quantisation via the tangent groupoid and new constraints on $D_{\mathrm{finite}}$ (the fermion mass matrix).
2 citations
TL;DR: In this article, the existence of adapted complex structures for real-analytic Riemannian manifolds is examined under the point view of complexifications of geodesic flows, and a sufficient criterion is given as to when a domain in the tangent bundle of the manifold is a maximal domain of definition of an adapted complex structure.
2 citations
01 Jan 2011
TL;DR: The Program for New Century Excellent Talents in Fujian Province University, China as mentioned in this paper was established by the Natural Science Foundation of China (NSCF) and the National Museum of Taiwan.
Abstract: Program for New Century Excellent Talents in Fujian Province University; Natural Science Foundation of China [10971170, 10601040]
2 citations