Betti and Tachibana Numbers

S. E. Stepanov

Matematičeskie zametki, Tome 95 (2014) no. 6, pp. 926-936

Résumé

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.

Détail

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