Semi-Riemannian Manifolds

In this chapter, we introduce the concept of a semi-Riemannian manifold and develop the necessary machinery to analyze its geometric structure. In section 2.1, we define the metric tensor on a manifold. Then in section 2.2, we introduce connections on a semi-Riemannian manifold compatible with the metric tensor. We also define parallel translation and geodesics determined by the semi-Riemannian structure of a manifold. In section 2.3 we study the curvatures of the Levi-Civita connection of a semi-Riemannian manifold. In section 2.4, we construct some differential operators on a semi-Riemannian manifold such as, divergence, gradient, Hessian tensor and Laplacian. Finally in section 2.5 we state the semi-Riemannian divergence theorem.