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Gromov’s convergence theorem and its application

Published online by Cambridge University Press:  22 January 2016

Atsushi Katsuda
Atsushi Katsuda*
Affiliation:
Department of Mathematics Faculty of Sciences Nagoya University, Nagoya 464, Japan
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One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 100 , December 1985, pp. 11 - 48
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

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