Consider $M$ to be a compact manifold and $f$ a morse function on $M$. Consider the unstable manifolds $W^{u}(x)$ defined using the flow of the negative gradient vector field of $f$. Let $c$ be a critical point of $f$, and $p\in W^{u}(c)$. Now let $v\in T_pM$ and we consider the point $\exp_{p}(v)\in M$. We know that $\exp_p(v)\in W^{u}(\tilde c)$ for some $\tilde c$ a critical point of $f$. I am interested in knowing if there is any condition that can put on $v$ such that $\exp_{p}(v)\in W^{u}(c)$? Maybe if we require that $v\in T_{p}W^{u}(c)$? And we could try to find a metric such that the unstable manifolds are totally geodesic, but I am not sure this is possible.
Does anyone know what kinda of conditions I can ask for $v$ so that I have $\exp_{p}(v)\in W^{u}(c)$?
Any insight is appreciated, thanks in advance.