Showing papers on "Unit tangent bundle published in 1982"

Factors of horocycle flows

Abstract: We classify up to an isomorphism all factors of the classical horocycle flow on the unit tangent bundle of a surface of constant negative curvature with finite volume.

Entropy estimates for geodesic flows

Abstract: Let M be a compact Riemannian manifold of (variable) negative curvature. Let h be the topological entropy and h μ the measure entropy for the geodesic flow on the unit tangent bundle to M . Estimates for h and h μ in terms of the ‘geometry’ of M are derived. Connections with and applications to other geometric questions are discussed.

Second-Order Tangent Structures

Abstract: Second-order differential processes have special significance for physics. Two reasonable generalizations of the procedure for constructing a tangent bundle over a smoothn-manifoldM yield different second-order structures, each projecting onto the standard first-order structureTM. One approach, based on the work of Ehresmann generalizes the notion of a tangent vector as a derivation. The other, based on the work of Yano and Ishihara generalizes the notion of a tangent vector as the velocity of a curve. The former leads toJ2M, the 2-jet vector bundle consisting of second-order derivations, the latter leads toT(2)M, the bundle of curves agreeing up to acceleration. Both project naturally ontoTM because the 1-jet bundle of first-order derivations and the bundle of curves agreeing up to velocity are isomorphs ofTM. Both generalizations admit extension to higher orders but the second-order case illustrates their differences and is important in applications. It is always true thatJ2M is a vector bundle; butT(2)M is a vector bundle if and only ifM has a linear connection and thenT(2)M≡TM⊕TM with fiber ℝ2n, whereasJ2M always has fiber\(\mathbb{R}^{(n^2 + 3n)/2} \). We compare these constructions and give some results aboutT(2)M and the principal bundleL(2)M to which it is associated. In a space-time there is a distinguished linear connection induced by the Lorentz metric, so both second-order tangent structures are available and the reduction ofJ2M toT(2)M is a considerable simplification in the casen=4. We show also that both second-order bundles have applications to the study of space-time boundaries.

The vortex: complex Hopf bundle and Morse theory

Abstract: The authors interpret the vortex solution of Nielsen-Olesen (1973) as a complex vector bundle associated to the second Hopf sphere bundle (analogously to consider the kink of the first Hopf bundle); the peculiarity of the soliton behaviour of the two-dimensional vortex stems from the non-trivial character of the fibration; the electromagnetic and scalar (Higgs) field are the connection and the section in this bundle respectively. Properties of this mathematical construction have their natural physical translation; for example the complex structure of the sphere S2 leads to a closed (Kahler) two-form, which has physical implications, and the fact that the vortex can be considered as the square root of the tangent bundle to the sphere implies a spinor nature for the vortex. A Morse theory of critical points suggests some Atiyah-Singer type of theorems, which have bearing on the stability of the multi-vortex solutions. They finish by a geometrical interpretation of the fractional charges found recently.