We develop a theory of tubular neighborhoods for the lower strata in manifold strati"ed spaces with two strata. In these topologically strati"ed spaces, manifold approximate "brations and teardrops play the role that "bre bundles and mapping cylinders play in smoothly strati"ed spaces. Applications include the classi"cation of neighborhood germs, a multiparameter isotopy extension theorem and a...

In this paper we show that a “locally Lipschitz” locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean topological manifold is a Lie group. This is a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem[1, 9] and also the Repovš-Ščepin Theorem [17] which holds ...

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications...

In the paper [SY80] it was shown that a Kähler manifold with strictly positive bisectional curvature was biholomorphic to CP. In this paper, we use the techniques developed by [SY80], to prove that a compact Kähler manifold with positive orthogonal bisectional curvature is biholomorphic to CP, a condition strictly weaker than positive bisectional curvature. This gives a direct elliptic proof of...

Contents Chapter 1. Introduction 7 1. Historical sketch 7 a. Homoclinic tangles 7 b. Geodesic flows 8 c. Picking up from Poincaré 8 d. Modern hyperbolic dynamics 9 e. The slowness of the initial development 9 2. Hyperbolic dynamics 10 3. Outline of this survey 10 a. Regularity 11 Chapter 2. Hyperbolic sets and stable manifolds 13 1. Definitions and examples 13 a. Hyperbolic linear maps, adapted...

We prove that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. This theorem was stated by Perelman and was used in his proof of the geometrization conjecture.

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M . This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric constr...

We consider Courant and Courant-Jacobi brackets on the stable tangent bundle TM ×R of a differentiable manifold and corresponding Dirac, Dirac-Jacobi and generalized complex structures. We prove that Dirac and Dirac-Jacobi structures on TM × R can be prolonged to TM × R, k > h, by means of commuting infinitesimal automorphisms. Some of the stable, generalized, complex structures are a natural g...

A famous theorem due to Nash ([3]) assures that every Riemannian manifold can be embedded isometrically into some Euclidean space E. An interesting question is whether for a complete manifold M we can find a closed isometric embedding. This note gives the affirmative answer to this question asked to the author by Paolo Piccione. In his famous 1956 article John Nash proved that every Riemannian ...

According to a result of one of the authors [7] there are at most a countable number of inequivalent differentiable actions of a compact Lie group on a compact differentiable manifold. The results above show that both the compactness of the manifold and the differentiability of the action are necessary assumptions. In the course of our proof we also prove the following theorem, which is an elem...