Bochner's technique in Einstein's non-symmetric geometry

TL;DR

This paper extends Bochner's technique to Einstein's non-symmetric geometry, defining key concepts, providing examples, and proving decomposition formulas and vanishing results for Laplacians.

Contribution

It introduces a framework for applying Bochner's technique to non-symmetric Einstein geometries, including new definitions, examples, and analytical results.

Findings

Established a Weitzenböck type decomposition formula.

Proved vanishing theorems for Laplacians in this setting.

Provided explicit examples of Einstein's connections with torsion.

Abstract

A. Einstein considered a manifold with a non-symmetric (0,2)-tensor $G=g+F$, where $g$ is a Riemannian metric and $F\ne0$, and a connection $\nabla$ with torsion $T$ such that $(\nabla_X G)(Y,Z)=-G(T(X,Y),Z)$. Guided by the almost Lie algebroid construction on a vector bundle, we define the basic concepts of Bochner's technique for Einstein's non-symmetric geometry, give a clear example of the Einstein's connection $\nabla$, prove Weitzenb\"{o}ck type decomposition formula and obtain vanishing results about the null space of the Bochner and Hodge type Laplacians.