Generally, the stable manifolds $W^s(p)$ of a diffeomorphism $\phi:M\to M$ is no embedded submanifold. The injective immersion $$ E^s:T_p^sM\to M $$ does not need to be a homeomorphism onto its image $W^s(p)$.
This is a popular image that illustrates this fact:
However, in the case that the diffeomorphism $\phi$ comes from a gradient flow, I think the stable manifold must be a submanifold. (More generally I think this is the case when there exists a strict Lyapunov function).
Can anybody tell me how to prove this?