Riemannian Exponential Map is a Homeomorphism outside the Cut Locus

Let $$M$$ be a connected and complete Riemannian manifold. The Hopf-Rinow Theorem guarantees that $$Exp_p$$, for any $$p \in M$$, is defined on all of $$T_p(M)$$. Further, this map is a diffeomorphism on a neighbourhood of the origin.

However, under the assumption of completeness and connectedness of $$M$$ is it a homeomorphism, is it a homeomorphism from $$M-C_p$$ to $$T_p(M)$$; where $$C_p$$ is the cut-locus of $$p$$?

Here I'll suppose $$M$$ is connected and complete.

Careful there is two notions of cut locus: the cut-locus at $$p$$ in $$T_pM$$ of a riemannian manifold is a subset of $$T_pM$$, not $$M$$ : it is the set of vectors $$v$$ in $$T_pM$$ for which $$\exp_p(tv)$$ is minimizing for $$t \in [0,1]$$ but is not minimizing on $$[0,1+\varepsilon]$$ for every $$\varepsilon >0$$. The cut locus at $$p$$ in $$M$$ is its image by $$\exp_p$$.

Consider the cut locus in $$T_pM$$, denoted by $$C_p$$. Let $$U = \{tv \mid t\in [0,1[, v \in C_p\}$$. Then $$U$$ is open and star-shaped. You can show that if $$tv \in U$$, then $$\mathrm{d}{\exp_p}_{tv} (w)$$ is the value of a certain Jacobi field at $$t$$, and thus is non-zero by asumptions if $$w$$ is non-zero. So $$\mathrm{d}\exp_{tv}$$ is invertible.

Okay right now we know that $$\exp : U \to \exp_p(U)$$ is smooth, has everywhere invertible differential, and by the very definition of $$U$$, is injective. Then it is a diffeomorphism as it is a injective local diffomorphism.

In fact, we have the disjoint decomposition $$M = \exp_p(U) \cup \exp_p(C_p)$$.

Remark there is no result here on all of $$T_pM$$ or on all of $$M$$ : we have shown that $$M$$ minus a closed subset (that can look really hideous) is diffeomorphic to a star shaped open subset of $$T_pM$$.

Edit : I think completeness is a strong hypothesis. We can do all this if $$M$$ has a pole $$p$$, that is a point $$p$$ for which the exponential map $$\exp_p$$ is surjective. Completeness says that everypoint is a pole.

• So $U=T_p(M)$ if and only if $M$ is non-positively curved?
– user683848
May 23 '20 at 9:58
• No : for example, for $M$ a genus $2$ Riemann surface with constant curvature $-1$, the exponential map is surjective by Hopf-RInow theorem as $M$ is compact, thus complete. But if $U=T_pM$, you would have that $\exp_p$ is a diffeomorphism of $U = T_pM$ onto its image which is $M$, but $U$ is non-compact and $M$ is. May 23 '20 at 10:01
• Right, but since any star-shaped domain is diffeomorphic to $T_p(M)$ then we can extend $\exp_p$ via some diffeomorphism $\phi:U\to T_p(M)$ such that $\phi\circ \exp_p:T_p \to \exp_p(U)$ is a diffeomorphism, right?
– user683848
May 23 '20 at 10:19
• Yes, but is it not anymore the exponential map May 23 '20 at 11:09