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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
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Abstract

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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

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