Vol. 78, No. 1, 1978

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Focal sets of submanifolds

Thomas E. Cecil and Patrick J. Ryan

Vol. 78 (1978), No. 1, 27–39

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 fλ factors through an immersion of the (n ν)-dimensional manifold Mλ into M. In this way, fλ(M) is an immersed submanifold of M.

Mathematical Subject Classification 2000
Primary: 53C40
Received: 2 May 1977
Revised: 15 February 1978
Published: 1 September 1978
Thomas E. Cecil
Department of Mathematics
College of the Holy Cross
Worcester MA 01610-2395
United States
Patrick J. Ryan