ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Tanaka, Minoru, Tokai University, & ITOH, Jin-ichi, Kumamoto University, Japan Sard Theorem for the distance function Let N be a closed submanifold of complete smooth Riemannian manifold M. A point q in M is called a critical point of the (Riemannian) distance function d(N, ·) from N if for any unit tangent vector v at q, there exists an N-segment γ joining q to N such that the angle made by v and γ at q is not greater than \frac{π}{2} . Here N-segment is a geodesic segment γ : [0, b] → M such that d(N, γ(t)) = t holds on [0, b]. A positive number c is called a critical value if there exists a critical point q of the distance function with d(N, q) = c. Theorem (Itoh-Tanaka). If M is 3-dimensional, then the set of all critical values of the distance function from N is of measure zero.