# Distance to cut locus and injectivity of the exponential map

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 $$d(p,q) and let $$\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 $$$$d(p,C_m(p))=sup\{r>0: \exp_p \text{is a diffeo on} B_r(p)\}.$$$$ Does this help?

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