The Weitzenböck Type Curvature Operator for Singular Distributions

Abstract

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.