Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T19:22:26.880Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on isometric immersions

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

C. Baikoussis  and
F. Brickell
C. Baikoussis
Affiliation:
Department of Mathematics University of IoanninaIoannina, Greece
F. Brickell
Affiliation:
Department of Mathematics University of SouthamptonSouthampton, England
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let N be a complete connected Riemannian manifold with sectional curvatures bounded from below. Let M be a complete simply connected Riemannian manifold with sectional curvatures KM(σ)≤ −a2 (a ≥ 0) and with dimension < 2 dim N. Suppose that N is isometrically immersed in M and that its image lies in a closed ball of radius ρ. Then sup(KN(σ) − KM(σ)) ≥ μ2(aρ)/ρ2 where the function μ is defined by μ(x) = x coth x for x > 0, μ(0) = 1 and the supremum is taken over all sections tangent to N.

MSC classification

Secondary: 53C40: Global submanifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 2 , October 1982 , pp. 162 - 166
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Baikoussis, C. and Koufogiorgos, T., ‘Isometric immersions of complete Riemannian manifolds into euclidean space’, Proc. Amer. Math. Soc. 79, (1980), 87–88.CrossRefGoogle Scholar
[2]Cheeger, J. and Ebin, D. G., Comparison theorems in Riemannian geometry (North-Holland, Amsterdam and Oxford, 1975).Google Scholar
[3]Ishihara, T., ‘Radii of immersed manifolds and nonexistence of immersions’, Proc. Amer. Math. Soc. 78 (1980), 276–279.Google Scholar
[4]Jacobowitz, H., ‘Isometric embedding of a compact Riemannian manifold into euclidean space’, Proc. Amer. Math. Soc. 40 (1973), 245–246.CrossRefGoogle Scholar
[5]Kobayashi, S. and Nomizu, N., Foundations of differential geometry, Vol. II (Interscience, New York, 1969).Google Scholar
[6]Moore, J. D., ‘An application of second variation to submanifold theory’, Duke Math. J. 42 (1975), 191–193.CrossRefGoogle Scholar
[7]Omori, H., ‘Isometric immersions of Riemannian manifolds’, J. Math. Soc. Japan 19 (1967), 205–214.Google Scholar