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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, a closed formula for the Chern character of the tangent bundles of the Hilbert schemes of points on a simply-connected surface was given, and the authors also gave a closed model for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface.
Abstract: We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface.
As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes.
We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface.
We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface.
4 citations
TL;DR: In this paper, the authors studied the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of a manifold and showed that flat n-tori, n ≥ 2, are focally rigid in the sense that if two flat tori are isometric up to rescaling.
Abstract: Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = S i 6 i called the focal decomposition of TM. The sets 6 i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n ≥ 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.
4 citations
TL;DR: In this article, the authors studied the symmetry transformations of a dynamical system defined on the tangent bundle of a Riemannian manifold. And they derived integrals quadratic in the velocities.
Abstract: The present paper is concerned with symmetry transformations of a dynamical system defined on the tangent bundle of a Riemannian manifold. Of present interest are infinitesimal symmetry transformations of the vector field which defines the dynamical system on the tangent bundle. It is known that a class of such transformations entails infinitesimal projective transformations leaving the vector field invariant. Symmetry algebras formed by such projective transformations are studied. It is shown which dynamical systems admit large symmetry algebras. As a result, two kinds of dynamical systems are determined, which have the base Riemannian manifolds of constant curvature with dimensions n?4. The systems are generalizations of the classical harmonic oscillator and Kepler problem usually considered in Euclidean spaces. First integrals quadratic in the velocities are obtained, which are also generalizations of the well‐known quadratic integrals for the above classical systems.
4 citations
4 citations
Journal Article•
TL;DR: In this article, the authors studied a one-to-one correspondence between DS 2 and a non cylindrical ruled surface and showed that for a restriction of an anti-symmetric linear vector field A along a spherical curve α(t) there exists a non-cylindrical rule surface which corresponds to α (t).
Abstract: T S 2 is a differentiable manifold of dimension 4. For every X ∈ T S 2 , if we set X = (p, x) we have < < p, x >= 0 since p is orthogonal to T p S 2 , therefore p = 1. Those there could exist a one-to-one correspondence between T S 2 and DS 2. In this paper we gave and studied a one-to-one correspondence among T S 2 , DS 2 and a non cylindrical ruled surface. We showed that for a restriction of an anti-symmetric linear vector field A along a spherical curve α(t) there exists a non-cylindrical ruled surface which corresponds to α(t) and has the following parametrization α(t, λ) = α(t) + A(α(t)) + λα(t) So it is possible to study non-cylindrical ruled surfaces as the set of (α(t), A(α(t))), where α(t) ∈ S 2 and A is an anti-symmetric linear vector field in R 3. Key words: dual unit sphere, non-cylindrical ruled surface, spherical curve, anti-symmetric linear vector field, tangent bundle 1. Anti-symmetric linear vector fields Let A = [a ij ] be a fixed real n × n matrix. For each such A we construct a vector field T A on R n by taking its value at each point x ∈ R n to be the negative of the result of applying the matrix A to the vector X, i.e. Definition 1. A vector field T A is called linear vector field ([3]). If A is an anti-symmetric (symmetric, orthogonal, etc.) matrix then T A is called an anti-symmetric (symmetric, orthogonal, etc.) linear vector field.
4 citations