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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 $p$?

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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.

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  • $\begingroup$ So $U=T_p(M)$ if and only if $M$ is non-positively curved? $\endgroup$
    – user683848
    May 23 '20 at 9:58
  • $\begingroup$ 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. $\endgroup$
    – Didier
    May 23 '20 at 10:01
  • $\begingroup$ 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? $\endgroup$
    – user683848
    May 23 '20 at 10:19
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    $\begingroup$ Yes, but is it not anymore the exponential map $\endgroup$
    – Didier
    May 23 '20 at 11:09

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