Showing papers on "Unit tangent bundle published in 1978"

Closed geodesics and the y-invariant

Abstract: In [3], Atiyah, Patodi and Singer introduced an invariant of a Riemannian manifold of dimension 4n 1. This invariant, which they called the p-invariant, is determined by the spectrum of a certain self-adjoint square root of the Laplacian on differential forms. It is non-local; that is, it is not obtained by integrating a universal polynomial in curvature over the manifold. Thus, unlike earlier invariants determined by the spectrum such as the Euler characteristic or the signature, it cannot be computed from the asymptotic expansion of Trace-t, the trace of the heat operator, as t goes to zero. However, recently, Colin de Verdiere [26], Chazarain [7] and Duistermaat-Guillemin [9] discovered a connection between the spectrum of the Laplacian and non-local information about a Riemannian manifold by studying the distribution trace of the fundamental solution of the wave equation. For generic manifolds the singularities of this distribution on the real line are at the set of lengths of closed geodesics and there is an asymptotic expansion at each singularity with coefficients giving information about the closed geodesic. It is an important question to decide if the 72-invariant can be determined from these data. The following formula shows this is indeed the case for manifolds of constant negative curvature and suggests a general formula. Let M be a compact oriented 4n 1 dimensional Riemannian manifold of constant negative curvature. Let $I be the set of primitive closed geodesics on M. Then each y C 9I determines the holonomy element R(7) e SO(4n 2) by parallel translation around y, the (linearized) Poincare map P(-i) C Sp (8n 4, R) and the length L('Y) of J. We stop to give a definition of P(7). Let q', denote the geodesic flow on S(M), the unit tangent bundle of M. Then a closed geodesic of length L corresponds to a fixed-point of pL. Then dpL maps the tangent space of that fixed-point to itself and preserves the geodesic flow direction and hence induces a transformation P(I) normal

$C^{3}$-actions and algebraic threefolds with ample tangent bundle

Abstract: One of the most challenging problems in complex differential geometry is the following conjecture of Frankel [3]. ( F-n) A compact Kaehler manifold M of dimension n with positive sectional (or more generally, positive holomorphic bisectional) curvature is biholomorphic to the complex projective space P n (C ).

On projective symmetries of dynamical systems

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.

Remarks on the use of the stable tangent bundle in the differential geometry and in the unified field theory

Abstract: © Gauthier-Villars, 1978, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.