Cut locus (Riemannian manifold)

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In Riemannian geometry, the cut locus of a point p in a manifold is roughly the set of all other points for which there are multiple geodesics connecting them from p. The cut locus is fundamental in certain analysis on manifolds, since the distance function from a point is a smooth function except on the cut locus.

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[edit] Definition

Fix a point p in a complete Riemannian manifold (M,g), and consider the tangent space TpM. It is a standard result that for sufficiently small v in TpM, the curve defined by γ(t) = expp(tv) for t belonging to the interval [0,1] is a minimizing geodesic (and is the unique minimizing geodesic connecting the two endpoints). (Note that expp denotes the exponential map from p.) The cut locus of p in the tangent space is defined to be the set of all vectors v in TpM such that γ(t) = expp(tv) is a minimizing geodesic for t \in [0,1] but is not minimizing for t \in [0,1 + \epsilon) for any ε > 0. The cut locus of p in M is defined to be image of the cut locus of p in the tangent space under the exponential map at p. Thus, we may interpret the cut locus of p in M as the points in the manifold where the geodesics beginning at p are no longer minimizing.

[edit] Characterization

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

[edit] Examples

On the standard round n-sphere, the cut locus 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.

[edit] Applications

The significance of the cut locus is that the distance function from a point p is smooth, except on the cut locus of p. In particular, it makes sense to take the gradient and Hessian of the distance function away from the cut locus. 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.

[edit] Notes

  1. ^ Petersen, Lemma 8.2.

[edit] References

Petersen, Peter. Riemannian Geometry, 1st ed. Springer-Verlag, 1998.