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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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14 citations
TL;DR: In this article, the authors developed homological conditions for a manifold to have a unit tangent bundle and applied these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the N-body problem.
Abstract: When a Hamiltonian system has a Kinetic + Potential structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure. We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the N-body problem. We show that the flow of the reduced planar N-body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.
14 citations
TL;DR: In this article, it was shown that the classical result of Riemannian geometry is in fact a simple consequence of a completely general theorem of the Calculus of Variations, which states that Jacobi fields along geodesics can be obtained as geodesies of the so-called complete lift of the metric g itself to the tangent bundle TQ.
Abstract: A classical result of Riemannian geometry states that Jacobi fields along geodesics of a Riemannian manifold (Q, g) can be obtained as geodesies of the so-called «complete lift» of the metric g itself to the tangent bundle TQ. We show that this classical result is in fact a very simple consequence of a completely general theorem of Calculus of Variations.
14 citations
01 Apr 1986
TL;DR: In this paper, Eisenbud et al. studied the relationship between the degrees of the line bundles of the splitting of O*Tp3 and the geometry of the curve C.
Abstract: Let O*Tp3 be the pull-back of the tangent bundle to P3 via a parametrization ik of a rational, reduced, irreducible curve C in P3 contained in an irreducible quadric surface. Since C is rational, the bundle i*Tp3 splits into the direct sum of three line bundles. In this paper we study the relationship between the degrees of the line bundles of the splitting of O*Tp3 and the geometry of the curve C. 0. Introduction. Let Tpr be the tangent bundle to the r-dimensional projective space pr over an algebraically closed field. Throughout this paper C will denote a rational, reduced, irreducible curve in pr of degree dc not contained in any hyperplane and 4: p' -l pr will be a parametrization of C. We will drop the subscript C anytime this will not lead to confusion. We consider the vector bundle 4'*Tpr on P' and, by abuse of language, we will refer to it as the tangent bundle restricted to the curve C. By a well-known theorem of Grothendieck V)*Tpr splits into the direct sum of r line bundles. The aim of this paper is to determine the decomposition of the tangent bundle restricted to a singular curve lying on a quadric surface in P3. The motivation for studying V)*Tpr comes from the relationship (suggested in [EV]) between it and the geometry of the embedded curve, and because of the information about the normal bundle, the Hilbert function, and the regularity of the curve one can derive from the splitting of 4'*Tpr. We distinguish between the cases where the quadric surface Q is singular from the ones where it is not. If C lies on a quadric cone then we will denote by a the intersection number of C with the lines of the ruling on Q outside the vertex; if C lies on a smooth quadric then (a, d a) will be its divisor class. In both cases we can assume that a < [d/2J. Our main results are the following THEOREM (0. 1). If C lies on a quadric cone, it passes through its vertex and one of the following holds: (i) a < Ld/3], (ii) Ld/3j < a and there exist two points on C of multiplicity a, (iii) Ld/3J < a and there exists a point on C of multiplicity a such that the total number of inflections at the point is a. Then O*Tpe v Opi (d + a) ED Opi (d + a) (D Op (2d 2a). Received by the editors November 4, 1985. 1980 Mathemotics Sukiect Cla&sifintion (1985 Revision). Primary 14H45. This paper is part of the author's Ph.D. thesis prepared under the supervision of Dr. D. Eisenbud (Brandeis University, May 1985). (g)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page
14 citations
TL;DR: In this paper, a generalization of the Einstein equation is presented by using the lifting of the Finsler metric to the tangent bundle, and a new definition of the equations of motion of the free particle in finsler space is presented.
Abstract: A Finslerian generalization of the Einstein equation is presented by using the lifting of the Finsler metric to the tangent bundle. A new definition of the equations of motion of the free particle in Finsler space is presented and it is shown that the motion of this particle resembles the motion of a spinning particle in curved space-time.
14 citations