Causal simplicity and (maximal) null pseudoconvexity

TL;DR

This paper explores the relationships between various pseudoconvexity properties in Lorentzian and Riemannian manifolds, providing examples that distinguish these properties in static spacetimes and their Riemannian counterparts.

Contribution

It introduces specific examples of spacetimes and manifolds that demonstrate the failure of certain pseudoconvexity properties despite other causality conditions being met.

Findings

Causally continuous and maximal null pseudoconvex spacetime can fail to be causally simple.

A Riemannian manifold can be minimally pseudoconvex but not convex.

Examples highlight nuanced differences between pseudoconvexity and convexity in geometric contexts.

Abstract

We consider pseudoconvexity properties in Lorentzian and Riemannian manifolds and their relationship in static spacetimes. We provide an example of a causally continuous and maximal null pseudoconvex spacetime that fails to be causally simple. Its Riemannian factor provides an analogous example of a manifold that is minimally pseudoconvex, but fails to be convex.