Showing papers on "Unit tangent bundle published in 1999"

Finslerian Fields in the Spacetime Tangent Bundle

Abstract: Finslerian fields are investigated in the arena of the maximal-acceleration invariantspacetime tangent bundle. A variety of differential-geometric Finslerian fields are exposited. Thestructure of Finslerian quantum fields receives particular emphasis. Also, possible generalizedactions are proposed for Finslerian strings and p-branes. 1999 Elsevier Science Ltd.

A Kaehler structure on the nonzero tangent bundle of a space form

Abstract: We obtain a Kaehler structure on the bundle of nonzero tangent vectors to a Riemannian manifold of constant positive sectional curvature This Kaehler structure is determined by a Lagrangian depending on the density energy only

Adapted complex structures and geometric quantization

Abstract: A compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure JA is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure JS is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of JA under an appropriate family of diffeomorphisms exists and agrees with JS .

Multiplication on the tangent bundle

Abstract: Manifolds with a commutative and associative multi- plication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability con- dition. They are studied here. They are closely related to discrim- inants and Lagrange maps. Frobenius manifolds are F-manifolds. As an application a conjecture of Dubrovin on Frobenius manifolds and Coxeter groups is proved.

$\mathbb Z^d$-covers of horosphere foliations

Abstract: Let $M$ be the unit tangent bundle of a compact manifold with negative sectional curvatures and let $\hat M$ be a $\mathbb Z^d$ cover for $M$. Let $\mu$ be the measure of maximal entropy for the associated geodesic flow on $M$ and let $\hat\mu$ be the lift of $\mu$ to $\hat M$. We show that the foliation $\hat{M^{s s}}$ is ergodic with respect to $\hat\mu$. (This was proved in the special case of surfaces by Babillot and Ledrappier by a different method.) Our method extends to certain Anosov and hyperbolic flows.

Hölder regularity of horocycle foliations

Abstract: Let M be a C∞, nonpositively curved manifold. A horosphere in M is the projection to M of a limit of metric spheres in the universal cover M (see §2). A horospherical foliation H is a foliation of the unit tangent bundle T 1M whose leaves consist of unit normal vector fields to horospheres.1 While regularity of horospherical foliations has been studied extensively for negatively curved manifolds M , considerably less is known in the nonpositively curved case. The most general result is due to P. Eberlein: if M is complete and nonpositively curved, then horospheres are C2, which implies that the individual leaves of H are C1. Further, the tangent distribution TH depends continuously the basepoint v ∈ T 1M (see [9]). Beyond Eberlein’s theorem, smoothness results have consisted mainly of counterexamples ([2],[5]); in particular, the best one could hope for in the case of a general compact, nonpositively curved M is for TH to be Holdercontinuous. In this paper we prove

Second order tangent vectors in riemannian geometry

Abstract: This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

On the vertical bundle of a pseudo-Finsler manifold

Abstract: We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable. Also, we find geometric properties of both leaves of Liouville distribution and the vertical distribution.

Eigenvalue pinching for Riemannian vector bundles

Abstract: Abstract We investigate some very general pinching results for eigensections with small eigenvalue of a Riemannian vector bundle. In particular, this gives pinching results for the eigenvalues of 1-forms on a compact Riemannian manifold, along with other applications.

Harmonic morphisms as unit normal bundles of minimal surfaces

Abstract: Let \(\) be an isometric immersion between Riemannian manifolds and \(\) be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space \(\) and necessary and sufficient conditions on f for the projection map \(\) to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres.

Bounds for Seshadri constants

Abstract: In the paper we present an alternative approach to the boundedness of Seshadri constants (which measure the local positivity) of nef and big line bundles at a general point of a complex--projective variety. Our approach is based on the construction of divisors in certain adjoint linear systems having isolated singularities of high order. A variant of the method gives bounds for Seshadri constants valid at arbitrary points of a smooth variety; these bounds, however, depend on the line bundle and the positivity of the tangent bundle of the variety in question.

Differential operators and the Frobenius pull back of the tangent bundle on G/P

Abstract: In this paper we study the sheaf of differential operators \(\) on a flag manifold X in characteristic p>0. We generalize the non-vanishing theorem of Haastert on the associated filtration of \(\). We make use of the lifting of X to characteristic zero and the complex geometry of X.

Nonrelativistic Geodesic Motion

Abstract: We show that any second-order dynamic equationon a configuration space X → R ofnonrelativistic time-dependent mechanics can be seen asa geodesic equation with respect to some (nonlinear)connection on the tangent bundle TX → X of relativisticvelocities. We compare relativistic and nonrelativisticgeodesic equations, and study the Jacobi vector fieldsalong nonrelativistic geodesics.

Tangent Bundle of a Manifold and its Homotopy Type

Abstract: There is a homotopy equivalence :M M between closed smooth manifolds of an odd dimension such that *TM , TM are stably isomorphic but not isomorphic to each other.

Density of periodic geodesics in the unit tangent bundle of a compact hyperbolic surface

Abstract: Let S be a compact oriented surface of constant curvature −1 and let T 1S be the unit tangent bundle of S endowed with the canonical (Sasaki) metric. We prove that T 1S has dense periodic geodesics, that is, the set of vectors tangent to periodic geodesics in T 1S is dense in TT 1S. Let M be a compact Riemannian manifold. M is said to have the DPG property (density of periodic geodesics) if the vectors tangent to periodic geodesics in M are dense in TM, the tangent bundle of M. A compact manifold is known to have this property if, for example, its geodesic flow is Anosov (see [1]), in particular if it is hyperbolic. In this note we prove that the unit tangent bundle of a compact oriented surface of constant curvature −1 shares with the surface the DPG property. ∗Partially supported by conicor, ciem (conicet) and secyt (unc). Mathematical Subject Classification: 53C 22, 53 C 30, 58F 17

Examples of Riemannian manifolds with positive curvature almost everywhere

Abstract: We show that the unit tangent bundle of S^4 and a real cohomology CP^3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature.