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.
Highlights
Distributions, as subbundles of the tangent bundle on a manifold, arise in such topics of mathematics and physics as fiber bundles, Lie groups actions, almost contact, Poisson and sub-Riemannian manifolds, e.g., [1,2,3,4,5]
Singularities play a crucial role in mathematics, and its applications in natural and technical sciences
There is definite interest of pure and applied mathematicians, e.g., [3,9], to singular distributions and foliations, i.e., having varying dimension: just mention Riemannian foliations, i.e., every geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets; an example is the orbital decomposition of the isometric actions of a Lie group
Summary
Distributions, as subbundles of the tangent bundle on a manifold, arise in such topics of mathematics and physics as fiber bundles, Lie groups actions, almost contact, Poisson and sub-Riemannian manifolds, e.g., [1,2,3,4,5]. A singular distribution D on a manifold M assigns to each point x ∈ M a linear subspace D x of the tangent space Tx M in such a way that, for any v ∈ D x , there exists a smooth vector field V defined in a neighborhood U of x and such that V ( x ) = v and V (y) ∈ Dy for all y of U. An image D = P( TM ) of a smooth endomorphism P ∈ End( TM) will be called a generalized vector subbundle of TM or a singular distribution.
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