On local characterization results in geometry and gravitation
pp. 541-570
in: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada (eds), From Riemann to differential geometry and relativity, Berlin, Springer, 2017Abstract
An important problem in differential geometry and in gravitation is to identify metrics in a fully coordinate independent manner. In fact, the very foundation of Riemannian geometry is based on the existence of a tensor, the Riemann or curvature tensor, which vanishes if and only if the metric is locally flat. Many other such local characterizations of metrics are known. The aim of this article is to present a brief selection of them as an example of the fruitful interplay between differential geometry and gravity.