Cut locus

In Riemannian geometry, the cut locus of a point ${\displaystyle p}$ in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from ${\displaystyle p}$, but it may contain additional points where the minimizing geodesic is unique, under certain circumstances. The distance function from p is a smooth function except at the point p itself and the cut locus.

Fix a point ${\displaystyle p}$ in a complete Riemannian manifold ${\displaystyle (M,g)}$, and consider the tangent space ${\displaystyle T_{p}M}$. It is a standard result that for sufficiently small ${\displaystyle v}$ in ${\displaystyle T_{p}M}$, the curve defined by the Riemannian exponential map, ${\displaystyle \gamma (t)=\exp _{p}(tv)}$ for ${\displaystyle t}$ belonging to the interval ${\displaystyle [0,1]}$ is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints. Here ${\displaystyle \exp _{p}}$ denotes the exponential map from ${\displaystyle p}$. The cut locus of ${\displaystyle p}$ in the tangent space is defined to be the set of all vectors ${\displaystyle v}$ in ${\displaystyle T_{p}M}$ such that ${\displaystyle \gamma (t)=\exp _{p}(tv)}$ is a minimizing geodesic for ${\displaystyle t\in [0,1]}$ but fails to be minimizing for ${\displaystyle t\in [0,1+\epsilon )}$ for each ${\displaystyle \epsilon >0}$. The cut locus of ${\displaystyle p}$ in ${\displaystyle M}$ is defined to be image of the cut locus of ${\displaystyle p}$ in the tangent space under the exponential map at ${\displaystyle p}$. Thus, we may interpret the cut locus of ${\displaystyle p}$ in ${\displaystyle M}$ as the points in the manifold where the geodesics starting at ${\displaystyle p}$ stop being minimizing.

...
Wikipedia