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

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 $\pi : (E, g) \to (B, -g_B)$ 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.

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Received 10 June 2017

Accepted 25 December 2018

Published 1 December 2021