Showing papers on "Unit tangent bundle published in 1972"

Geodesic flow in certain manifolds without conjugate points

Abstract: A complete simply connected Riemannian manifold H without conju- gate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic y of H tends uniformly to zero as the distance from p to y tends uniformly to infinity. A complete manifold Mis a uniform Visibility manifold if it has no conju- gate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and Tt the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to Tt then Tt is topologically transitive on SM. We also prove that if M' is a normal covering of Mthen Tt is topologically transitive on SM' if Tt is topologically transitive on SM. Introduction. Much research, past and present, has been devoted to proving the topological transitivity of the geodesic flow in manifolds with curvature K^O that satisfy various conditions, usually including compactness. An essential part of all the proofs is the fact that if Af is a complete manifold with K^O and simply connected covering space H then any two distinct points of H are joined by a unique geodesic. Manifolds Af without conjugate points are characterized by this property in the simply connected covering space H and it is natural to ask what further conditions on Af are sufficient to imply that the geodesic flow is topo- logically transitive. Results have been obtained (mostly in the compact case) by Green, Hedlund, Klingenberg, Morse and others under the assumption that the manifold Af admits another (closely related) metric g* with curvature K= — 1 or #gc<0. In this paper we extend and unify many of these results by requiring that Af be a uniform Visibility manifold (defined above and more precisely in §1). To comple- ment the main results stated above we obtain the following criteria for the existence of uniform Visibility manifolds.

Tangent bundles of homogeneous spaces are homogeneous spaces

Abstract: In this paper we describe how the tangent bundle of a homogeneous space can be viewed as a homogeneous space. The purpose of this note is to establish a simple result on the structure of the tangent bundle of a homogeneous space. Even though it is both natural and elementary it does not appear to be in the literature. We shall associate with every Lie group G another Lie group G*, constructed as a semidirect product of G with the Lie algebra of G (the precise definition is given below). Our result is: THEOREM. If a Lie group G acts transitively and with maximal rank on a differentiable manifold X, then G* acts transitively and vith maximal rank on the tangent bundle of X. Clearly, our result implies that the tangent bundle of a coset space G/H is again a coset space and moreover, is of the form] G*IK for some closed subgroup K of G*. We will compute K below. We now define G * and prove the theorem. Let L be the Lie algebra of G, thought of as the tangent space of G at the identity. For each g E G, we let ad(g) denote the differential at the identity of the inner automorphism x-gg-1 of G. Thus ad is a (not necessarily one-to-one) homomorphism of G into the group of linear automorphisms of L. We define G* as the product manifold L x G, with the group operation given by (1) (a, g) (a', g') = (a + ad(g)(a'), gg'). The verification that G* is a group is trivial and will be omitted. Also, it is clear that the operation defined by (1) is differentiable, so that G* is a Lie group. Now, let G act differentiably on a manifold X. For each x E X, let OX: G--X be defined by Ox(g)=gx. If x E X, then the differential of Ox at the identity maps L into XZ (the tangent space of X at x). If a E L, we let d(x)=dO.(a). It is easy to see that a is a smooth vector field on X. Use T(X) to denote the tangent bundle of X. Received by the editors November 24, 1971. AMS 1970 suibject classifications. Primary 53C30, 57E25.

Tensor fields and connections on a cross-section in the tangent bundle of order r

Abstract: Let M be an ^-dimensional differentiable manifold and Tr(M) the tangent bundle of order r over M, ri^l being an integer [1], [3], [4]. The prolongations of tensor fields and connections given in the differentiable manifold M to its tangent bundle of order r have been studied in [1J, [2], [3] [4], [7], [8] and [9]. If V is a vector field given in M, V determines a cross-section in Tr(M). For the cases r=l and r=2, Yano [7] and Tani [5] have studied, on the cross-section determined by a vector field F, the behavior of the prolongations of tensor fields and connections in M to T(M) (i.e., Ά(M)) and Γa(M), respectively. The purpose of this paper is to study, on the cross-section determined by a vector field V, the behavior of the prolongations of these geometric objects in M to Tr(M) ( r^ l ) . In §1 we summarize the results and properties we need concerning the prolongations of tensor fields and connections in M to Tr(M). Proofs of the statements in §1 can be found in [1], [2], [3], [4] and [8]. In § 2 we study the cross-section determined in Tr(M) by a given vector field V in M In § 3 we study the behavior of prolongations of tensor fields on the cross-section. In §4 we study the prolongations of connections given in M to Tr{M) along the cross-section and some of their properties. We assume in the squel that the manifolds, functions, tensor fields and connections under consideration are all of differentiability of class C°°. Several kinds of indices are used as follows: The indices λ, μ, v, • ••, s, t, u, ••• run through the range 0,1, 2, ••• r; the indices h, i, j , k, m, ••• run through the range 1, 2, ••• n. Double indices like {v)h are used, where O^i^r, l^h^n. The indices Λ, B, C, ••• run through the range (1)1, (1)2, •••, (l)n, (2)1, •••, (2)n, •••, (r)l, •••, (r)n. For a given function / on M, the notation / ( 0 ) is sometimes substituted by f° for simplicity. Summation notation Σ J β l with respect to h> i, j , k, m, ••• (=1, 2, ••• n) is omitted while summation notation with respect to λ, μ, v, •••, s, t, u •••, from 0 to r, will be kept. For example,

The Tangent Bundle of a Topological Manifold

Abstract: (1972). The Tangent Bundle of a Topological Manifold. The American Mathematical Monthly: Vol. 79, No. 10, pp. 1090-1096.

Some Cross-Section Theorems on the Tangent Bundle over a Finslerian Manifold (*)(**).

Abstract: Let T(M) be the tangent bundle over a Finslerian manifold M of n-dimension endowed with the Cartan connection ∇. One makes T(M) into a 2n dimensional affinely connected manifold by assigning a connection ∇c to T(M). The cross-section\(\mathfrak{B}\) of a vector field V defined in M reveals in T(M) an n-dimensional submanifold and its geometry is developed by means of the affine subspace theory and of the affine collineations in the base Finsler manifold.

Homology tangent bundles and universal bundles

Abstract: We find results about the evaluation map from the group of homeomorphisms of a closed manifold M and also about fibre bundles where M is the fibre. These facts follow from the observation that the homology tangent bundle is induced from a universal bundle pair.

The tangent microbundle of a suitable manifold

Abstract: The purpose of this note is to generalize to the topological category the fact that a suitable differentiable manifold is parallelizable (Theorem 4 of [1]). This result has a "folk-theorem" status in some quarters, but I believe that in view of the recent interest in H-manifolds [2], it would be desirable to have the result on record. Let M be an n-manifold. Define A: M--M x M to be the diagonal map, and 7T', 7T2: M x M--M to be the projections on the first and second factor respectively. Milnor [3] calls the diagram A: M;?M x M: Il the tangent microbundle of M, where for each point beM there exists an open set Ub in M containing b, an open set Vb in Mx M containing A(b), and a homeomorphism h: Vb--Ub x Rn such that the following diagram commutes: Ub idxO A lUb Ub x R-< h V