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Open AccessArticle

On the Betti and Tachibana Numbers of Compact Einstein Manifolds

1
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
2
Department of Mathematics, Russian Institute for Scientific and Technical Information of the Russian Academy of Sciences, 20, Usievicha Street, 125190 Moscow, Russia
3
Department of Data Analysis and Financial Technologies, Finance University, 49-55, Leningradsky Prospect, 125468 Moscow, Russia
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1210; https://doi.org/10.3390/math7121210
Received: 23 November 2019 / Revised: 6 December 2019 / Accepted: 7 December 2019 / Published: 9 December 2019
(This article belongs to the Special Issue Inequalities in Geometry and Applications)
Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold ( M , g ) with an Einstein constant α > 0 is a homological sphere when the minimum of its sectional curvatures > α / ( n + 2 ) ; in particular, ( M , g ) is a spherical space form when the minimum of its sectional curvatures > α / n . Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold with α < 0 . View Full-Text
Keywords: Einstein manifold; sectional curvature; Betti number; Tachibana number; spherical space form Einstein manifold; sectional curvature; Betti number; Tachibana number; spherical space form
MDPI and ACS Style

Rovenski, V.; Stepanov, S.; Tsyganok, I. On the Betti and Tachibana Numbers of Compact Einstein Manifolds. Mathematics 2019, 7, 1210.

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