# On the fundamental group of semi-Riemannian manifolds with positive curvature tensor

## TL;DR

This paper explores how certain positive curvature conditions in semi-Riemannian manifolds influence the finiteness of their fundamental groups, providing theoretical results and a counterexample.

## Contribution

It establishes conditions under which the fundamental group of the fiber is finite in semi-Riemannian manifolds with positive curvature, including new examples with non-integrable horizontal distributions.

## Key findings

- Finiteness of the fundamental group of the fiber under positive curvature conditions.
- Existence of semi-Riemannian submersions with positive curvature and non-integrable horizontal distribution.
- Theoretical link between curvature positivity and topological finiteness in semi-Riemannian geometry.

## Abstract

This paper presents an investigation of the relation between some positivity of the curvature and the finiteness of fundamental groups in semi-Riemannian geometry. We consider semi-Riemannian submersions $\pi : (E, g) \rightarrow (B, -g_{B}) $ under the condition with $(B, g_{B})$ Riemannian, the fiber closed Riemannian, and the horizontal distribution integrable. Then we prove that, if the lightlike geodesically complete or timelike geodesically complete semi-Riemannian manifold $E$ has some positivity of curvature, then the fundamental group of the fiber is finite. Moreover we construct an example of semi-Riemannian submersions with some positivity of curvature, non-integrable horizontal distribution, and the finiteness of the fundamental group of the fiber.