Comparison of two formulas for metric connections in the bundle of Dirac spinors

TL;DR

This paper compares two explicit formulas for metric connections in the Dirac spinor bundle, proves their equivalence, and derives a formula linking spinor curvature with Riemann curvature.

Contribution

It introduces and proves the equivalence of two formulas for metric connections in Dirac spinor bundles and relates spinor curvature to Riemann curvature.

Findings

Proved the equivalence of two explicit formulas for metric connections.

Derived a formula relating spinor curvature tensor to Riemann curvature tensor.

Enhanced understanding of geometric structures in Dirac spinor bundles.

Abstract

Two explicit formulas for metric connections in the bundle of Dirac spinors are studied. Their equivalence is proved. The explicit formula relating the spinor curvature tensor with the Riemann curvature tensor is rederived.