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We study generalizations of Lorentzian warped products with one-dimensionalbase of the form $I\times_f X$, where $I$ is an interval, $X$ is a length spaceand $f$ is a positive continuous function. These generalized cones furnish animportant class of Lorentzian length spaces in the sense of [Kunzinger,S\"amann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimalcausality properties that allow for explicit descriptions of all underlyingnotions. In addition, synthetic sectional curvature bounds of generalized conesare directly related to metric curvature bounds of the fiber $X$. The interestin such spaces comes both from metric geometry and from General Relativity,where warped products underlie important cosmological models (FLRW spacetimes).Moreover, we prove singularity theorems for these spaces, showing thatnon-positive lower timelike curvature bounds imply the existence of incompletetimelike geodesics.

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems