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We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

Abstract. The purpose of this paper is to define the r-th Tachibana number t r of an n-dimensional closed and oriented Riemannian manifold (M, g) as the dimension of the space of all conformal Killing r-forms for r = 1, 2, . . . , n − 1 and to formulate some properties of these numbers as an analogue to properties of the r-th Betty number b r of a closed and oriented Riemannian manifold. Mathematics Subject Classification (2000): 58A10, 58G25Key words: compact Riemannian manifold, differential form, kernel of elliptic operator, scalar invariant.0. Introduction 0.1. In the present paper we define Tachibana numbers of a compact and oriented Riemannian manifold and basing on the definition explain well-known results, including our own. This paper contains a new view point on well-known facts of the theory of Riemannian manifolds. Moreover in our paper we used ideas that were reflected in classic monograph [24].The paper is based on the author's report given at the International Conference "Differential Equations and Topology" dedicated to the Centennial Anniversary of L.S. Pontryagin, Moscow, June 17-22, 2008 (see [12]). 0.2. We shall concerned with four subspaces of the vector space Ω r (M, R) of exterior differential r-forms on an n-dimensional closed and oriented Riemannian manifold (M, g), namely, the subspace H r (M, R) of harmonic differential r-forms, the subspace T r (M, R) of conformal Killing differential r-forms, the subspace K r (M, R) of co-closed conformalKilling r-forms and the subspace P r (M, R) of closed conformal Killing r-forms for any r = 1, 2, . . . , n − 1.For a closed and orientable (M, g), we shall call the number t r = dimT r (M, R) -the Tachibana number, k r = dimK r (M, R) -the Killing number and p r = dimP r (M, R) -the planar number of (M, g). We shall show that the Tachibana number t r are conformal 1