Cut locus

Remarks

Example: on the unit sphere $S^n$, every unit-speed geodesic from $p$ is a great circle with $d(p, \gamma_v(t)) = t$ for $t \le \pi$ and $d(p, \gamma_v(t)) < t$ just after $t = \pi$. So $t_c(v) = \pi$ for every $v$, and the cut locus of $p$ is the single point $-p$.

Non-example: in $\R^n$ with the Euclidean metric, $d(p, p + tv) = t$ for all $t > 0$, so $t_c(v) = \infty$ for all $v$, there are no cut points, and the cut locus is empty.

Let $q = \gamma_v(t_c(v))$ be the cut point of $p$ along $\gamma_v$. Then $q$ is the first point conjugate to $p$ along $\gamma_v$, or there are two distinct minimizing geodesics from $p$ to $q$, or both. The cut locus is a closed set of measure zero, and $\exp_p$ maps the open star-shaped set $\set{tv : \abs{v} = 1, \ 0 \le t < t_c(v)}$ diffeomorphically onto $M$ minus the cut locus of $p$.