The Lipschitz Continuity of the Distance Function to the Cut Locus

Let N be a closed submanifold of a complete smooth Riemannian manifold M and Uν the total space of the unit normal bundle of N . For each v ∈ Uν, let ρ(v) denote the distance from N to the cut point of N on the geodesic γv with the velocity vector γ̇v(0) = v. The continuity of the function ρ on Uν is well known. In this paper we prove that ρ is locally Lipschitz on which ρ is bounded; in particular, if M and N are compact, then ρ is globally Lipschitz on Uν. Therefore, the canonical interior metric δ may be introduced on each connected component of the cut locus of N, and this metric space becomes a locally compact and complete length space. Let N be an immersed submanifold of a complete C∞ Riemannian manifold M and π : Uν → N the unit normal bundle of N . For each positive integer k and vector v ∈ Uν, let a number λk(v) denote the parameter value of γv, where γv denotes the geodesic for which the velocity vector is v at t = 0, such that γv(λk(v)) is the k-th focal point (conjugate point for the case in which N is a point) of N along γv, counted with focal (or conjugate) multiplicities. A unit speed geodesic segment γ : [0, a]→M emanating from N is called an N -segment if t = d(N, γ(t)) on [0, a]. By the first variation formula, an N -segment is orthogonal to N . A point γv(t0) on an N -segment γv, v ∈ Uν, is called a cut point of N if there is no N -segment properly containing γ[0, t0]. For each v ∈ Uν, let ρ(v) denote the distance from N to the cut point on γv of N. Whitehead [27] investigated the structure of the conjugate locus and the cut locus of a point on a real analytic Finsler manifold. He determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood (cf. also [25]) and proved the continuity of the function ρ. In compact symmetric spaces, T. Sakai [19] and M. Takeuchi [23] determined the detailed structure of the cut locus of a point. The detailed structure of the cut locus of a point in a 2-dimensional Riemannian manifold has been investigated by Poincaré, Myers, and others [7], [11], [13]. Hartman first tried to show the absolute continuity of the function ρ when M is 2-dimensional. He proved in [8] that if ρ is of bounded variation, then ρ is absolutely continuous. Recently, Hebda [11] and the first named author [13] independently proved Ambrose’s problem by showing that ρ is absolutely continuous on a closed arc on which ρ is bounded when N is a point in a 2-dimensional Riemannian manifold. Therefore, the cut locus of a point in a compact 2-dimensional Received by the editors October 14, 1998 and, in revised form, April 13, 1999. 2000 Mathematics Subject Classification. Primary 53C22; Secondary 28A78. Supported in part by a Grant-in-Aid for Scientific Research from The Ministry of Education, Science, Sports and Culture, Japan. c ©2000 American Mathematical Society