Differential Geometry and Tensors

This chapter is on modern differential geometry and tensors. This topic is treated in modern rigorous language involving differentiable manifolds, tangent vectors, forms, higher rank tensors as elements of tensor products of tangent and cotangent spaces, the affine connection, Riemannian metric, curvature, geodesics, and geodesic deviation. The chapter ends with a brief discussion on vector bundles. The chapter lays the foundation for understanding global methods in relativity, such as black hole horizons, singularities, gauge theories, etc., in particle physics.