Showing papers on "Unit tangent bundle published in 1995"

Tangent lifts of Poisson and related structures

Abstract: The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector fields. These tangent lifts are studied with application to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

The tangent bundle of a Calabi-Yau manifold---deformations and restriction to rational curves

Abstract: The tangent bundle ℐX of a Calabi-Yau threefoldX is the only known example of a stable bundle with non-trivial restriction to any rational curve onX. By deforming the direct sum of ℐX and the trivial line bundle one can try to obtain new examples. We use algebro-geometric techniques to derive results in this direction. The relation to the finiteness of rational curves on Calabi-Yau threefolds is discussed.

Applications of Dynamics to Compact Manifolds of Negative Curvature

Abstract: Consider closed Riemannian manifolds with negative sectional curvature. There are three natural dynamics associated with the Riemannian structure: the geodesic flow on the unit tangent bundle, the dynamics of the invariant foliations of the geodesic flow, and the Brownian motion on the universal cover of the manifold. These dynamics define global asymptotic objects such as growth rates or measures at infinity. For locally symmetric negatively curved spaces, these objects are easy to compute and to describe. In this paper, we survey some of their properties and relations in the general case.

Homogeneous spaces of nonpositive curvature and their geodesic flow

Abstract: Consider the geodesic flow on the unit tangent bundle SH of a 1-connected, irreducible homogeneous space H of nonpositive curvature. We prove that any flow invariant, isometry invariant C0-function on SH is necessarily constant, unless H is symmetric of higher rank. As the main applications, we obtain rigidity and partial classification results for spaces H whose geodesic symmetries are (asymptotically) volume-preserving.

A horizontal Lévy process on the bundle of orthonormal frames over a complete riemannian manifold

Abstract: We establish an existence theorem for a class of SDE’s driven by Levy processes on a manifold. As an application we consider an SDE driven by horizontal vector fields on the orthonormal frame bundle of a Riemannian manifold. The canonical projection of the solution of this equation on to the base is considered as a candidate for a “Levy process on a Riemannian manifold”.

Liftings of $1$-forms to the linear $r$-tangent bundle

Abstract: Let r; n be xed natural numbers. We prove that for n-manifolds the set of all linear natural operators T ! T T (r) is a nitely dimensional vector space over R. We construct explicitly the bases of the vector spaces. As a corollary we nd all linear natural operators T ! T r. All manifolds and maps are assumed to be innnitely diierentiable. 0. Let r; n be xed natural numbers. Given a manifold M we denote by T r M = J r (M; R) 0 the space of all r-jets of maps M ! R with target 0. This is a vector bundle over M with the source projection. The dual vector bundle (T r M) of T r M is denoted by T (r) M and called the linear r-tangent bundle of M. We denote the bre of T r M and T (r) M over x by T r x M and T (r) x M respectively. Every embedding ' : M ! N of two n-manifolds induces a vector bundle homomorphism T r ' : T r M ! T r N over ' deened by T r '(j r x) = j r '(x) (' ?1) for any : M ! R and any x 2 M with (x) = 0, where by j r x we denote the r-jet of at x. This embedding induces also a vector bundle homomorphism T (r) ' : T (r) M ! T (r) N over ' dual to T r ' ?1 , i.e. T (r) '(()(j r '(x)) = (j r x (')) for any 2 T (r) x M, any x 2 M and any j r '(x) 2 T r '(x) N, cf. 4]. In this paper we study the problem how a 1-form ! on a manifold M can induce a 1-form on T (r) M and a section of T r M ! M. This problem is reeected in the concept of linear natural operators T ! T T (r) and T ! T r , cf. 4]. In the fundamental monograph 4] there is a very general deenition of natural operators. We restrict ourselves to the following one. Deenition 0.1. Let r; n be xed natural numbers. Let 1 (M) denotes the vector space of all 1-forms on M and ?T r M denotes the vector space of all …

Compact Hermitian surfaces of pointwise constant holomorphic sectional curvature

Abstract: Let M = ( M, J, g ) be an almost Hermitian manifold and U ( M )the unit tangent bundle of M . Then the holomorphic sectional curvature H = H ( x ) can be regarded as a differentiable function on U ( M ). If the function H is constant along each fibre, then M is called a space of pointwise constant holomorphic sectional curvature. Especially, if H is constant on the whole U ( M ), then M is called a space of constant holomorphic sectional curvature. An almost Hermitian manifold with an integrable almost complex structure is called a Hermitian manifold. A real 4-dimensional Hermitian manifold is called a Hermitian surface. Hermitian surfaces of pointwise constant holomorphic sectional curvature have been studied by several authors (cf. [2], [3], [5], [6] and so on).

Holomorphic sectional curvature of the tangent bundle

Abstract: In order to investigate the differential structure of a Riemannian manifold (M, g), it seems a powerful tool to study the differential structure of its tangent bundle TM. In this point of view, K. Aso [1] studied, using the Sasaki metric $\tilde{g}$, the relation between the curvature tensor on (M, g) and that on (TM, $\tilde{g}$).

Natural Transformations of Lagrangians into p-Forms on the Tangent Bundle

Abstract: This paper presents without proofs some theorems giving a complete characterization of natural transformations of finite order of Lagrangians into p-forms on the tangent bundle over n-dimensional manifolds for n ≥ p + 1 (except for the case p = 0, n = 1).

The tangent sphere bundle of a surface in terms of the ricci tensor

Abstract: We investigate the tangent sphere bundle of a 2-dimensional Riemannian manifold M with the natural Riemannian structure g in the two classes, given by A.Gray([3]), including Ricci-parallel Riemannian manifolds. Also, we prove that is conformally flat if and only if is locally symmetric. The motivation of this paper are a fact that a 3-dimensional Ricci-parallel Riemannian manifold is locally symmetric and a result([2]) that the natural Riemannian structure of the tangent sphere bundle of a Riemannian manifold is locally symmetric if and only if either the baes manifold is flat or is of dimension 2 and constant curvature 1.