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Isometric Immersions of almost Hermitian Manifolds

Published online by Cambridge University Press:  20 November 2018

Alfred Gray
Alfred Gray*
Affiliation:
University of California, Berkeley, California University of Maryland, College Park, Maryland
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The Lefschetz theorem on hyperplane sections, as proved by Andreotti and Frankel (1), depends upon the following result.

THEOREM. If M is a non-singular affine algebraic variety of real dimension 2k of complex n-space, then

This theorem, which is interesting in itself, has been strengthened by Milnor (7), who showed that M has the homotopy type of a k-dimensional CW-complex.

In this paper we generalize the above theorem in two directions. First, we replace complex n-space by some other complete simply connected Riemannian manifold which either has non-positive curvature or is a compact symmetric space. Secondly, we allow M and to be quasi-Kâhlerian (see below) instead of Kählerian.

We first introduce some notation. Let M and be C∞ Riemannian manifolds with M isometrically immersed in . Denote by 〈, 〉 the metric tensor of either M or .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 456 - 459
Copyright
Copyright © Canadian Mathematical Society 1969

References

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