On the Betti and Tachibana Numbers of Compact Einstein Manifolds

Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok
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Vladimir Rovenski: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
Sergey Stepanov: Department of Mathematics, Russian Institute for Scientific and Technical Information of the Russian Academy of Sciences, 20, Usievicha Street, 125190 Moscow, Russia
Irina Tsyganok: Department of Data Analysis and Financial Technologies, Finance University, 49-55, Leningradsky Prospect, 125468 Moscow, Russia

Mathematics, 2019, vol. 7, issue 12, 1-6

Abstract: 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 .

Keywords: Einstein manifold; sectional curvature; Betti number; Tachibana number; spherical space form (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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