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On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold

Published online by Cambridge University Press:  20 January 2009

Pui-Fai Leung
Pui-Fai Leung
Affiliation:
Department of Mathematics, National University of Singapore, Kent Ridge, Singapore, 0511
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Let Mn be an n-dimensional smooth compact Riemannian manifold. By a theorem of Nash, we can think of it as an isometrically immersed submanifold in some higher dimensional Euclidean space ℝn+m. Viewing in this way we can compare the intrinsic geometry of M to its extrinsic geometry. Classically, the Gauss equation

where K(X,Y) denotes the sectional curvature in M corresponding to the plane spanned by the two orthonormal vectors X, Y and B denotes the second fundamental form gives one of the most important relations between the intrinsic and extrinsic geometries of M. In this note we shall prove the following.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 3 , October 1985 , pp. 305 - 311
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

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