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 Mat. Zametki, 2014, Volume 95, Issue 6, Pages 926–936 (Mi mz9369)

Betti and Tachibana Numbers

S. E. Stepanov

Financial University under the Government of the Russian Federation, Moscow

Abstract: The Tachibana numbers $t_r(M)$, the Killing numbers $k_r(M)$, and the planarity numbers $p_r(M)$ are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing $r$-forms with $1\le r\le n-1$ “globally” defined on a compact Riemannian $n$-manifold $(M,g)$, $n\ge 2$. Their relationship with the Betti numbers $b_r(M)$ is investigated. In particular, it is proved that if $b_r(M)=0$, then the corresponding Tachibana number has the form $t_r(M)=k_r(M)+p_r(M)$ for $t_r(M)>k_r(M)>0$. In the special case where $b_1(M)=0$ and $t_1(M)>k_1(M)>0$, the manifold $(M,g)$ is conformally diffeomorphic to the Euclidean sphere.

Keywords: compact manifold, Tachibana number, Killing number, planarity number, Betti number, conformal Killing form, conformal Killing (co)closed form.

DOI: https://doi.org/10.4213/mzm9369

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English version:
Mathematical Notes, 2014, 95:6, 856–864

Bibliographic databases:

UDC: 514.764.2
Revised: 11.03.2013

Citation: S. E. Stepanov, “Betti and Tachibana Numbers”, Mat. Zametki, 95:6 (2014), 926–936; Math. Notes, 95:6 (2014), 856–864

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz9369
• https://doi.org/10.4213/mzm9369
• http://mi.mathnet.ru/eng/mz/v95/i6/p926

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This publication is cited in the following articles:
1. Stepanov S.E. Tsyganok I.I. Mikes J., “Overview and Comparative Analysis of the Properties of the Hodge-de Rham and Tachibana Operators”, Filomat, 29:10 (2015), 2429–2436
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