Cut locus (Riemannian manifold)

In Riemannian geometry, the cut locus of a point [math]\displaystyle{ p }[/math] in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from [math]\displaystyle{ p }[/math], 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.

Definition

Fix a point [math]\displaystyle{ p }[/math] in a complete Riemannian manifold [math]\displaystyle{ (M,g) }[/math], and consider the tangent space [math]\displaystyle{ T_pM }[/math]. It is a standard result that for sufficiently small [math]\displaystyle{ v }[/math] in [math]\displaystyle{ T_p M }[/math], the curve defined by the Riemannian exponential map, [math]\displaystyle{ \gamma(t) = \exp_p(tv) }[/math] for [math]\displaystyle{ t }[/math] belonging to the interval [math]\displaystyle{ [0,1] }[/math] is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints. Here [math]\displaystyle{ \exp_p }[/math] denotes the exponential map from [math]\displaystyle{ p }[/math]. The cut locus of [math]\displaystyle{ p }[/math] in the tangent space is defined to be the set of all vectors [math]\displaystyle{ v }[/math] in [math]\displaystyle{ T_pM }[/math] such that [math]\displaystyle{ \gamma(t)=\exp_p(tv) }[/math] is a minimizing geodesic for [math]\displaystyle{ t \in [0,1] }[/math] but fails to be minimizing for [math]\displaystyle{ t = 1 + \varepsilon }[/math] for every [math]\displaystyle{ \varepsilon \gt 0 }[/math]. The cut locus of [math]\displaystyle{ p }[/math] in [math]\displaystyle{ M }[/math] is defined to be image of the cut locus of [math]\displaystyle{ p }[/math] in the tangent space under the exponential map at [math]\displaystyle{ p }[/math]. Thus, we may interpret the cut locus of [math]\displaystyle{ p }[/math] in [math]\displaystyle{ M }[/math] as the points in the manifold where the geodesics starting at [math]\displaystyle{ p }[/math] 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 [math]\displaystyle{ q }[/math] is in the cut locus of [math]\displaystyle{ p }[/math] in [math]\displaystyle{ M }[/math]. A standard result is that either (1) there is more than one minimizing geodesic joining [math]\displaystyle{ p }[/math] to [math]\displaystyle{ q }[/math], or (2) [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] are conjugate along some geodesic which joins them. It is possible for both (1) and (2) to hold.

Examples

Cut locus C(P) of a point P on the surface of a cylinder. A point Q in the cut locus is shown with two distinct shortest paths [math]\displaystyle{ \gamma_1, \gamma_2 }[/math] connecting it to P.

On the standard round n-sphere, the cut locus of a point consists of the single point opposite of it (i.e., the antipodal point). On an infinitely long cylinder, the cut locus of a point consists of the line opposite the point.

Applications

The significance of the cut locus is that the distance function from a point [math]\displaystyle{ p }[/math] is smooth, except on the cut locus of [math]\displaystyle{ p }[/math] and [math]\displaystyle{ p }[/math] itself. In particular, it makes sense to take the gradient and Hessian of the distance function away from the cut locus and [math]\displaystyle{ p }[/math]. 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.

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.

See also

• Caustic (mathematics)