Focal sets of submanifolds

Thomas E. Cecil and Patrick J. Ryan

Vol. 78 (1978), No. 1, 27–39

DOI: 10.2140/pjm.1978.78.27

Abstract

This is a study of the manifold structure of the focal set of an immersed submanifold in a real space form M. A typical result is the following:

Theorem. Let M be an orientable (immersed) hypersurface in M which is complete with respect to the induced metric. Let λ be a differentiable principal curvature of constant multiplicity ν > 1 on M. Then the focal map f_λ factors through an immersion of the (n − ν)-dimensional manifold M_λ into M. In this way, f_λ(M) is an immersed submanifold of M.

Mathematical Subject Classification 2000

Primary: 53C40

Milestones

Received: 2 May 1977

Revised: 15 February 1978

Published: 1 September 1978