# How can I show that the map of cut point is continuous?

For a complete riemannian manifold $$M$$ with a point $$p_0\in M$$, we have already kbow that the exponential map $$exp_{p_0}(v):\mathbb{S}^{m-1}\subset T_{p_0}M\rightarrow M$$ is well defined. Meanwhile we can define $$\mu_{p_0}(v)=\sup\left\{s>0:d(exp_{p_0}(sv),p_0)=s\right\}$$. Clearly we see that $$exp_{p_0}(\mu_{p_0}(v)v)$$ is the cut point on the geodesic $$exp_{p_0}(tv)$$ for all $$v\in \mathbb{S}^{m-1}\subset T_{p_0}M$$. I want to show that the map $$\mu_{p_0}:\mathbb{S}^{m-1}\rightarrow \mathbb{R}_+$$ is continuous. I have tried to use some statements about conjugate points and Jacobi field, but I cannot complete the proof. Can anyone give me some hints about it ? References about such statement are also good.

• Here is a paper which shows that the distance is not only continuous, but in fact uniformly Lipschitz for compact manifolds. However, I believe it's not possible to do much better than that for a generic metric, since cut loci can be quite irregular. jstor.org/stable/221962 Jun 21, 2021 at 17:18
• I should point out that $\mu$ need not be finite, and as such maps into $\mathbf{R} \cup \{ \infty \}$. Jun 23, 2021 at 16:29