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

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.