Let $M$ be a Riemannian manifold, $p \in M$ and $C_m(p)$ shall denote the cut locus of $p$. In his „Riemannian Geometry“ Do Carmo says that $\exp_p$ is injective on a open ball $B_r(p)$ of radius $r$ if and only if $r\leq d(p,C_m(p))$. I have however trouble proving the if direction.

He says that this is a consequence of the fact that if $q \notin C_m(p)$, then there is unique minimizing geodesic from $p$ to $q$. He also proved the fact that if $\gamma(t_0)$ is the cut point of $p=\gamma(0)$ along $\gamma$, then either $\gamma(t_0)$ is the first conjugate point of $p$ along $\gamma$ or there exists a different geodesic of equal length connecting $p$ and $\gamma(t_0)$.

As I said I have trouble proving that if $\exp_p$ is injective on $B_r(p)$, then $r\leq d(p,C_m(p)).$ My attempt: Assume $r>d(p,C_m(p)).$ Choose a point $q \in C_m(p)$ with $d(p,q)<r$ and let $\gamma$ denote a minimizing geodesic joining $p$ and $q$. If there is a different geodesic of same length joining these two points, then $\exp_p$ is not inhective on $B_r(p).$ But I can‘t come up with a contradiction when $q$ is the first conjugate point of $p$ along $\gamma.$

I would appreciate a hint on how to proceed or a proof.

I was able to prove \begin{equation} d(p,C_m(p))=sup\{r>0: \exp_p \text{is a diffeo on} B_r(p)\}. \end{equation} Does this help?

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    $\begingroup$ The key fact is that no geodesic is minimizing past its first conjugate point. Does do Carmo prove that? $\endgroup$ – Jack Lee Nov 15 '18 at 2:44
  • $\begingroup$ Thanks, this helped. $\endgroup$ – Frieder Jäckel Nov 15 '18 at 7:49

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