What does the "m" stand for in cut locus $C_m(p)$?

Question

I'm reading do Carmo "Riemannian geometry" chapter 13, where cut locus is defined. While other authors use ${\rm cut}(p)$ for the cut locus of $p\in M$, do Carmo denotes it by $C_m(p)$. I wonder what exactly the $m$ stands for in this notation. Perhaps it stands for the manifold $M$, but since the $m$ appears in lower case all the way, I doubt that it is due to some typo.

Answer 1

Possibly, the author meant to write $C_M(p)$ but the typesetter used $m$ for some reason. The most common notation for the cut locus of a point $p$ in a Riemannian manifold $M$ is $C(p)$ or $Cut(p)$. Some authors use $m$ to denote a point in $M$ (for instance, in the book of Cheeger and Ebin) and then the cut locus is denoted $C(m)$. However, sometimes $C(p)$ is used to denote the conjugate locus of $p$, so maybe the subscript $m$ was used to distinguish the two. In any case, this is just a notation.