On the Bochner technique for singular distributions

TL;DR

This paper extends Bochner's technique to singular and regular distributions with statistical structures on manifolds, deriving new formulas and vanishing theorems for tensor Laplacians.

Contribution

It generalizes Bochner's method to distributions with statistical structures, introducing a modified connection, exterior derivative, and curvature operator.

Findings

Derived a Bochner-Weitzenbock type formula for distributions.

Established vanishing theorems for the null space of the Hodge Laplacian.

Extended the theory to singular and regular distributions with statistical structures.

Abstract

In this paper we continue our recent study of a manifold endowed with a singular or regular distribution, determined as the image of the tangent bundle under a smooth endomorphism, and generalize Bochner's technique to the case of a distribution with a statistical type structure. Following the theory of statistical structures on Riemannian manifolds and construction of an almost Lie algebroid on a vector bundle, we define the modified statistical connection and exterior derivative on tensors. Then we introduce the Weitzenbock type curvature operator on tensors and derive the Bochner-Weitzenbock type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian on a distribution.