The spinor bundle of Riemannian products

TL;DR

This paper investigates the relationship between the spinor bundle of a Riemannian product manifold and the spinor bundles of its factors, showing an isomorphism under certain metric conditions without holonomy restrictions.

Contribution

It establishes an explicit isomorphism between the spinor bundle of a Riemannian product and tensor products of the factor bundles for a specific class of metrics, without holonomy assumptions.

Findings

Spinor bundle of product manifold can be expressed as a tensor product of factor bundles.

Isomorphism holds for a special class of metrics on the product manifold.

No holonomy conditions are required for the main result.

Abstract

In this note we compare the spinor bundle of a Riemannian manifold $(M=M_1\times...\times M_N,g)$ with the spinor bundles of the Riemannian factors $(M_i,g_i)$. We show, that - without any holonomy conditions - the spinor bundle of $(M,g)$ for a special class of metrics is isomorphic to a bundle obtained by tensoring the spinor bundles of $(M_i,g_i)$ in an appropriate way.