Basic Differential Geometry

This first chapter recapitulates the basic concepts of differential geometry that are used throughout the book. This encompasses differentiable manifolds, tensor fields, affine connections, metric tensors, pseudo-Riemannian manifolds, Levi–Civita connections, curvature tensors and Lie derivatives. The dimension of the manifold and the signature of the metric are kept general so that the results can be subsequently applied either to the whole spacetime or to some submanifold of it.