Tachibana, Killing and planarity numbers of compact Riemannian manifolds

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We present definitions and properties of conformal Killing, Killing and planarity forms on a Riemannian manifold and determine Tachibana, Killing and planarity numbers as an analog of the well known Betti numbers. We state some set of conditions to characterize these numbers. Moreover, we formulate the main results on the relationship between the Betti, Tachibana, Killing and planarity numbers.