Impact of curves on Lorentzian submersions

This study investigates the role of curve theory in characterizing Lorentzian submersions, which fall into the category of semi-Riemannian submersions. The survey begins with a classification that examines the causal character of the horizontal and vertical components of a curve transferred from the total manifold to the base manifold along Lorentzian submersion. Taking into account this classification during the transfer of curves along a Lorentzian submersion, the study then concentrates on certain types of curves along a Lorentzian submersion, including the non-null Frenet curve, non-null circle, non-null helix, and a geodesic. Through detailed analysis, it explores how these curves are influenced by a Lorentzian submersion and the geometric meanings that arise from this interaction. A significant focus of the study is on exploring the transformation of a Cartan-framed lightlike curve from the total manifold to the base manifold via a Lorentzian submersion. Inspired by foundational work in differential geometry and the theory of Riemannian submersions, particularly those of O’Neill (1996) and Gray (1967), this work combines and improves Ikawa’s attempt (1985) to describe some curves such as circles and helices in an indefinite Riemannian manifold. It reveals novel findings into the application of these theories to Lorentzian submersions.