The Cut Locus and Geodesic Completeness

The cut locus captures where geodesics from a point cease to be globally minimizing. Together with geodesic completeness, it governs the large-scale geometry of Riemannian manifolds.

The Cut Locus

Definition8.7Cut point and cut locus

Let $\gamma$ be a unit-speed geodesic starting at $p$ . The cut point of $p$ along $\gamma$ is the point $\gamma(t_{\mathrm{cut}})$ where $\gamma$ ceases to be a minimizing geodesic:

t_{\mathrm{cut}} = \sup\{t > 0 : d(p, \gamma(t)) = t\}.

The cut locus $\mathrm{Cut}(p)$ is the set of all cut points along all geodesics from $p$ .

Theorem8.4Characterization of cut points

A point $q = \gamma(t_0)$ is the cut point of $p$ along $\gamma$ if and only if at least one of the following holds:

$q$ is the first conjugate point of $p$ along $\gamma$ .
There exists another minimizing geodesic from $p$ to $q$ distinct from $\gamma$ .

ExampleCut loci of model spaces

$S^n$ : $\mathrm{Cut}(N) = \{S\}$ (the antipodal point). Every geodesic from the north pole is minimizing until it reaches the south pole.
Flat torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ : $\mathrm{Cut}(0) = \{(x,y) : x = 1/2 \text{ or } y = 1/2\}$ , forming a square. There are no conjugate points; the cut locus arises from multiple minimizing geodesics.
$\mathbb{R}^n$ : $\mathrm{Cut}(p) = \emptyset$ (every geodesic is minimizing for all time).
$\mathbb{R}P^n$ : $\mathrm{Cut}(p) \cong \mathbb{R}P^{n-1}$ (an $(n-1)$ -dimensional projective space).

Structure of the Cut Locus

Theorem8.5Properties of the cut locus

For a complete Riemannian manifold:

$M \setminus \mathrm{Cut}(p) = \exp_p(U)$ where $U \subset T_pM$ is an open star-shaped domain, and $\exp_p|_U$ is a diffeomorphism (so $M \setminus \mathrm{Cut}(p)$ is diffeomorphic to an open subset of $\mathbb{R}^n$ ).
$\mathrm{Cut}(p)$ has measure zero in $M$ .
If $M$ is compact, $\mathrm{Cut}(p)$ is a deformation retract of $M \setminus \{p\}$ and has the homotopy type of a CW-complex of dimension $\leq n-1$ .

Geodesic Completeness

Definition8.8Geodesic completeness

A Riemannian manifold is geodesically complete if every geodesic can be extended to all of $\mathbb{R}$ , or equivalently (by Hopf-Rinow), if the metric space $(M, d)$ is complete.

RemarkCompleteness and compactness

Every compact Riemannian manifold is complete (bounded closed sets are compact). The converse is false: $\mathbb{R}^n$ and $\mathbb{H}^n$ are complete but not compact. However, completeness combined with curvature bounds can imply compactness (e.g., Bonnet-Myers theorem: positive Ricci curvature lower bound implies bounded diameter, hence compactness).

ExampleIncomplete manifolds

The punctured plane $\mathbb{R}^2 \setminus \{0\}$ with the Euclidean metric is incomplete: geodesics heading toward the origin cannot be extended. Adding a conformal factor can also create incompleteness: the metric $g = e^{-1/r^2}(dx^2 + dy^2)$ on $\mathbb{R}^2$ may cause geodesics to reach "infinity" in finite time.