ABSTRACT
This chapter discusses semi-Riemannian manifolds. It shows that null cones determine the metric up to a conformai factor for metrics which are not definite. The chapter considers sectional curvature. This curvature is related to tidal accelerations using the Jacobi equation. The chapter explains generic condition corresponds to a tidal acceleration assumption and Einstein equations. The Einstein equations thus link geometry in terms of the metric and curvature to physics in terms of the distribution of mass and energy. The curvature tensor, Ricci curvature, scalar curvature, and sectional curvature may all be calculated in local coordinates using the metric tensor components and the first two partial derivatives of these components. Thus, the metric tensor determines the curvatures. The physical motivation for studying Lorentzian manifolds is the assumption that a gravitational field may be effectively modeled by some Lorentzian metric g defined on a suitable four-dimensional manifold.
