The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this chapter. General coordinate transformations, tangent spaces, vectors and tensors are described. Lie derivatives and covariant derivatives are motivated and defined. The concepts of parallel transport and a connection is introduced and the relation between the Levi-Civita connection and geodesics is elucidated. Christoffel symbols the Riemann tensor are defined as well as the Ricci tensor, the Ricci scalar and the Einstein tensor, and their algebraic and differential properties are described (though technical details of the derivationa of the Rimeann tensor are let to an appendix).

This last part of the book introduces the Einstein equation – the basic equation of general relativity, in much the same way that Maxwell’s equations are the basic equations of electromagnetism. Geometries such as the Schwarzschild geometry, or those of the FRW cosmological models, are particular solutions of the Einstein equation. Just three new mathematical ideas are needed to give an efficient and standard discussion of the Einstein equation: a more precise definition of vectors in terms of directional derivatives; the notion of dual vectors as a linear map from vectors to real numbers; and the covariant derivative of a vector field in curved spacetime. These mathematical concepts are introduced in this chapter.