Splitting theorems in presence of an irrotational vector field

TL;DR

This paper introduces new splitting theorems for semi-Riemannian manifolds with irrotational vector fields, leading to various decompositions and applications in Lorentzian geometry.

Contribution

It develops novel splitting theorems for manifolds with irrotational vector fields, extending previous results to include twisted, warped, and direct decompositions.

Findings

Derived conditions for manifold decompositions based on irrotational vector fields

Established applications to Lorentzian manifolds

Analyzed $ extbf{S}^{1} imes L$ type decompositions

Abstract

New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can get twisted, warped or direct decompositions. Some applications to Lorentzian manifold are shown and also $\mathbf{S}^{1}\times L$ type decomposition is treated.