Differentiable Manifolds

Abstract

In this first chapter of the purely mathematical part on the most important tools of differential geometry needed for GR, we introduce several basic concepts connected with the notion of a differentiable manifold. We give two definitions of a differentiable manifold. The standard one starts with a topological space. One can alternatively begin with a set and introduce the topology with a given atlas. This approach is not only practical to construct differentiable manifolds, but is also more appropriate from a physical point of view. On the basis of this notion, we introduce differentiable maps between differentiable manifolds (shortly manifolds in what follows), immersions, embeddings, and submanifolds.