ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Itoh, Jin-ichi, Kumamoto University, Tanaka, Minoru, Tokai University, Japan The Lipschitz continuity of the distance function to the cut locus Let M be a complete smooth Riemannian manifold and S_{p}M be the unit tangent space of p ∈ M. For each positive integer k and vector v ∈ S_{p}M, let a number λ_{k}(v) denote the parameter value of γ_{v}, where γ_{v} denotes the geodesic frpm p with its initial direction v, such that γ_{v}(λ_{k}(v)) is the k-th conjugate point along γ_{v} with counting conjugate multiplicities. For each v ∈ S_{p}M, let ρ(v) denote the distance from p to the cut point along γ_{v}. Theorem A. For each positive integer k and v ∈ S_{p}M with λ_{k}(v) < ∞, λ_{k} is locally Lipschitz around v. Theorem B. For each v ∈ S_{p}M with ρ(v) < ∞, ρ is locally Lipschitz around v. In particular, if M is compact, then ρ is globally Lipschitz on S_{p}M.