Contractibility of manifolds by means of stochastic flows

Abstract

In the paper [Probab. Theory Relat. Fields, 100 (1994) 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold $M$ under which the fundamental group $\pi_1 M=0$. We prove that in fact under the similar conditions the manifold $M$ is contractible, that is all homotopy groups $\pi_k M$, $k\geq1$, vanish.