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Total curvature of manifolds in self-immersed manifolds

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Abstract

Chern-Lashof [3] and Kuiper [5] showed the total absolute curvature of a manifold in Euclidean space equals the mean value of the number of critical points of height functions. Teufel [10] proved that a similar result holds for the total absolute curvature of a manifold in a unit sphere. The purpose of this paper is to extend Teufel's result to a relation between the total absolute curvature of some manifolds in self-immersed manifolds and the mean value of the number of zeros of certain vector fields.

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  1. Mathematical Department of Tokushima University, Josanjima, 770, Tokushima, Japan

    Toru Ishihara

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  1. Toru Ishihara
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Ishihara, T. Total curvature of manifolds in self-immersed manifolds. Manuscripta Math 39, 201–218 (1982). https://doi.org/10.1007/BF01165785

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  • DOI: https://doi.org/10.1007/BF01165785

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