Cut time in a unit tangent direction

Assume countable choice, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $(M,g)$ be a complete, connected, boundaryless Riemannian manifold, let $p\in M$, and let $v\in T_{p}M$ be a unit vector. Hopf–Rinow (Hopf–Rinow theorem) makes the radial geodesic ${\gamma }_v(t)={\exp }_p(tv)$ defined for every $t\in \mathbb{R}$. Its cut time is

$$c_p(v):=\sup \{t>0:d_g(p,{\exp }_p(tv))=t\}\in (0,+\mathrm{\infty }]\mathrm{.}$$

We write $c(v)$ when $p$ is fixed. The set in the supremum is nonempty: Sufficiently short geodesic segments are uniquely minimizing gives equality for every sufficiently small $t>0$. The value $+\mathrm{\infty }$ is allowed when every positive radial segment minimizes. This definition alone does not assert that a finite supremum is attained.

For a zero-dimensional manifold the unit sphere in each tangent space is empty, so there are no directions on which to evaluate $c_p$. In dimension one the two unit directions, when present, are treated separately by the same formula. The only choice assumption here is the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ carried by the cited global and local geodesic results; fixing one $p$ and $v$ makes no use of a choice function on a family of directions and does not invoke full AC.