Affine connections on singular warped products

TL;DR

This paper develops new geometric tools called semi-symmetric Koszul forms on singular warped products, expressing their curvature in terms of factor manifolds, and extends these results to multiply warped products.

Contribution

It introduces semi-symmetric Koszul forms on singular semi-Riemannian manifolds and generalizes curvature expressions to singular multiply warped products.

Findings

Expressed curvature of semi-symmetric Koszul forms in terms of factor manifolds.

Extended results to singular multiply warped products.

Connected Koszul forms to almost product structures.

Abstract

In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated to the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated to the almost product structure and their curvature of semi-regular almost product warped products are expressed in terms of those of the factor manifolds. Furthermore, we generalize the results in \cite{St2} to singular multiply warped products.