Betti and Tachibana numbers

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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 ≤ r ≤ n − 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 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.