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Distance functions and umbilic submanifolds of hyperbolic space

Published online by Cambridge University Press:  22 January 2016

Thomas E. Cecil  and
Patrick J. Ryan
Thomas E. Cecil
Affiliation:
Department of MathematicsCollege of the Holy Cross
Patrick J. Ryan
Affiliation:
Department of MathematicsIndiana University at South Bend
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In 1972, Nomizu and Rodriguez [5] found the following characterization of the complete umbilic submanifolds of Euclidean space.

Theorem A. Let Mn, n ≥ 2, be a connected, complete Riemannian manifold isometrically immersed in a Euclidean space Em. Every Morse function of the form Lp has index 0 or n at all of its critical points if and only if Mnis embedded as a Euclidean n-subspace or a Euclidean n-sphere in Em.

Keywords

53C4053B25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 74 , July 1979 , pp. 67 - 75
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

[1] Cartan, E., Leçons sur la géométrie des espaces de Riemann, deuxième édition, Gauthier-Villars, Paris, 1946.Google Scholar
[2] Cecil, T., A characterization of metric spheres in hyperbolic space by Morse theory, Tohoku Math. J. 26 (1974), 341–351.CrossRefGoogle Scholar
[3] Cecil, T., Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math. J. 55 (1974), 5–31.Google Scholar
[4] Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Vo. II, John Wiley and Sons, Inc., New York, 1969.Google Scholar
[5] Nomizu, K. and Rodriguez, L., Umbilical submanifolds and Morse functions, Nagoya Math. J. 48 (1972), 197–201.Google Scholar