Cut point and cut locus of a point

Assume countable choice as carried through the declared dependencies. Let $(M,g)$ be a complete, connected, boundaryless Riemannian manifold and $p\in M$. Write $S_{p}M=\{v\in T_{p}M:\mathrm{\mid }v{\mathrm{\mid }}_g=1\}\mathrm{.}$ For each $v\in S_{p}M$, let $c_p(v)$ be the cut time from Cut time in a unit tangent direction and put ${\gamma }_v(t)={\exp }_p(tv)$. When $c_p(v)<\mathrm{\infty }$, the point $q={\gamma }_v(c_p(v))={\exp }_p(c_p(v)v)$ is the cut point of $p$ along ${\gamma }_v$. The cut locus of $p$ is $\mathrm{Cut}(p)=\{{\exp }_p(c_p(v)v):v\in S_{p}M,c_p(v)<\mathrm{\infty }\}\mathrm{.}$

This is the finite-supremum definition used by Lee, Chapter 10, in the section “Geodesics Do Not Minimize Past Conjugate Points” (PDF label P206, printed p.190). The endpoint is indeed the last minimizing point on its specified ray: Minimizing along a geodesic is an initial interval property gives minimization at a finite cut time and initial-interval behavior. If $c_p(v)$ is finite, every $0\le t<c_p(v)$ is also minimizing: the supremum property gives a minimizing time larger than $t$, and initial-interval behavior then applies. No $t>c_p(v)$ is minimizing, since $c_p(v)$ is an upper bound for the minimizing times. If $c_p(v)=+\mathrm{\infty }$, every finite radial segment minimizes and this direction contributes no point to $\mathrm{Cut}(p)$.

If $M$ is empty there is no base point $p$, so there is no set $\mathrm{Cut}(p)$ to evaluate. In dimension zero, $S_{p}M=\mathrm{\varnothing }$ and hence $\mathrm{Cut}(p)=\mathrm{\varnothing }$. In dimension one, the two unit tangent directions at each point are handled separately by the same formula. The definition retains the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption; forming this set makes no simultaneous choice of directions and adds no further choice principle.