Decomposition of stochastic flows and rotation matrix

We provide geometrical conditions on the manifold for the existence of the Liao's factorization of stochastic flows \cite{ML}. If $M$ is simply connected and has constant curvature then this decomposition holds for any stochastic flow, conversely, if every flow on $M$ has this decomposition then $M$ has constant curvature. Under certain conditions, it is possible to go further on the factorization: $ \varphi_t = \xi_t \circ \Psi_t \circ \Theta_t$, where $\xi_t$ and $\Psi_t$ have the same properties of Liao's decomposition and $(\xi_t \circ \Psi_t)$ are affine transformations on $M$. We study the asymptotic behaviour of the isometric component $\xi_t$ via rotation matrix, providing a Furstenberg-Khasminskii formula for this skew-symmetric matrix.