Betti and Tachibana numbers (Q2342047)

The author investigates a relationship among the Tachibana numbers \(t_r(M)\), the Killing numbers \(k_r(M)\), and the planarity numbers \(p_r(M)\) on a Riemannian manifold \(M\), depending on the Betti numbers \(b_r(M)\). The main result is 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. There are also other relations obtained when certain assumptions on the Ricci tensor are posed.