Communications in Analysis and Geometry

Volume 29 (2021)

Number 5

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

Pages: 1255 – 1277

DOI:  https://dx.doi.org/10.4310/CAG.2021.v29.n5.a8

Author

Jun-Ichi Mukuno (Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Japan)

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

Received 10 June 2017

Accepted 25 December 2018

Published 1 December 2021