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The diameter of an immersed Riemannian manifold with bounded mean curvature

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Th. Koufogiorgos  and
Ch. Baikoussis
Th. Koufogiorgos
Affiliation:
Department of Mathematics, University of Ioannina, Ioannina, Greece
Ch. Baikoussis
Affiliation:
Department of Mathematics, University of Ioannina, Ioannina, Greece
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Abstract

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Let M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

MSC classification

Secondary: 53C40: Global submanifolds

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 2 , August 1981 , pp. 189 - 192
Copyright
Copyright © Australian Mathematical Society 1982

References

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Omori, H. (1967), ‘Isometric immersions of Riemannian manifold’, J. Math. Soc. Japan 19, 205–214.CrossRefGoogle Scholar