Abstract
In this paper, we extend our previous study regarding a Riemannian manifold endowed with a singular (or regular) distribution, generalizing Bochner’s technique and a statistical structure. Following the construction of an almost Lie algebroid, we define the central concept of the paper: The Weitzenböck type curvature operator on tensors, prove the Bochner–Weitzenböck type formula and obtain some vanishing results about the null space of the Hodge type Laplacian on a distribution.
Highlights
IntroductionDistributions (subbundles of the tangent bundle) on a manifold are used to build up notions of integrability, and of a foliation, e.g., [1,2,3]
Distributions on a manifold are used to build up notions of integrability, and of a foliation, e.g., [1,2,3]
In Example 4 in [14] we showed that (21)(a) is reasonable: div( PX ) − (div P) = 0 with P = f f ∗ holds for an f -structure with parallelizable kernel if and only if both distributions f ( TM) → Λ2 ( (TM)) and ker f are harmonic
Summary
Distributions (subbundles of the tangent bundle) on a manifold are used to build up notions of integrability, and of a foliation, e.g., [1,2,3]. There is definite interest of pure and applied mathematicians to singular distributions and foliations, i.e., having varying dimension, e.g., [4,5]. Another popular mathematical concept is a statistical structure, i.e., a Riemannian manifold endowed e such that the tensor ∇. We generalize Bochner’s technique to a Riemannian manifold endowed with a singular (or regular) distribution and a statistical type connection, continue our study [13,14,15,16,17,18]. The assumptions that we use are reasonable, as illustrated by examples
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