Some Curvature Problems in Semi-Riemannian Geometry

TL;DR

This survey reviews key results on the curvature of semi-Riemannian metrics, focusing on estimates of the Riemann tensor in asymptotically flat manifolds and constructing Lorentzian metrics satisfying the dominant energy condition, motivated by the positive mass theorem.

Contribution

It consolidates existing results on semi-Riemannian curvature estimates and metric constructions related to the positive mass theorem in a comprehensive survey.

Findings

Estimates of the Riemann tensor in asymptotically flat manifolds

Construction methods for Lorentzian metrics satisfying energy conditions

Connections between curvature estimates and the positive mass theorem

Abstract

In this survey article we review several results on the curvature of semi-Riemannian metrics which are motivated by the positive mass theorem. The main themes are estimates of the Riemann tensor of an asymptotically flat manifold and the construction of Lorentzian metrics which satisfy the dominant energy condition.