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 under similar analytical conditions the manifold $M$ is contractible, that is all homotopy groups $\pi_n M$, $n\geq1$, vanish.
"Contractibility of manifolds by means of stochastic flows." Topol. Methods Nonlinear Anal. 52 (2) 599 - 611, 2018. https://doi.org/10.12775/TMNA.2018.022