Cut locus

Formal definition

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 ${\displaystyle t=1+\epsilon }$ for every ${\displaystyle \epsilon >0}$. Thus the cut locus in the tangent space is the boundary of the set $${\displaystyle \{v\in T_{p}M|d({\exp }_{p}v,p)=\Vert v\Vert \}}$$ where ${\displaystyle d}$ denotes the length metric of ${\displaystyle M}$, and ${\displaystyle \Vert \cdot \Vert }$ is the Euclidean norm of ${\displaystyle T_{p}M}$. 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.

The least distance from p to the cut locus is the injectivity radius at p. On the open ball of this radius, the exponential map at p is a diffeomorphism from the tangent space to the manifold, and this is the largest such radius. The global injectivity radius is defined to be the infimum of the injectivity radius at p, over all points of the manifold.

Characterization

Suppose ${\displaystyle q}$ is in the cut locus of ${\displaystyle p}$ in ${\displaystyle M}$. A standard result is that either (1) there is more than one minimizing geodesic joining ${\displaystyle p}$ to ${\displaystyle q}$, or (2) ${\displaystyle p}$ and ${\displaystyle q}$ are conjugate along some geodesic which joins them. It is possible for both (1) and (2) to hold.

Applications

The significance of the cut locus is that the distance function from a point ${\displaystyle p}$ is smooth, except on the cut locus of ${\displaystyle p}$ and ${\displaystyle p}$ itself. In particular, it makes sense to take the gradient and Hessian of the distance function away from the cut locus and ${\displaystyle p}$. This idea is used in the local Laplacian comparison theorem and the local Hessian comparison theorem. These are used in the proof of the local version of the Toponogov theorem, and many other important theorems in Riemannian geometry.

For the metric space of surface distances on a convex polyhedron, cutting the polyhedron along the cut locus produces a shape that can be unfolded flat into a plane, the source unfolding. The unfolding process can be performed continuously, as a blooming of the polyhedron. Analogous methods of cutting along the cut locus can be used to unfold higher-dimensional convex polyhedra as well.

Cut locus of a subset

One can similarly define the cut locus of a submanifold of the Riemannian manifold, in terms of its normal exponential map.