A differentiable manifold, defined with the help of collections of charts, comes with basic notions of calculus prior to the introduction of the metric. The most important is how one may define vectors as directional derivatives, 1-forms via their dual pair, and build up general tensors from these two. Partial derivatives for functions extend to the Lie derivative, while a special subclass of tensors known as differential forms admits another type of derivative known as the exterior derivative. We develop calculus based on these most basic differential structures, ending with the Stokes theorem. Once a differentiable manifold is given, one can turn to a further structure of a metric and its accompanying Levi-Civita connection, leading to the Riemann curvature tensor. Numerous additional structures made possible by the covariantly constant metric are introduced, such as raising and lowering of indices, Killing vector fields, the volume form, the Hodge star map, geodesics, and geodesic normal coordinates.

Once a differentiable manifold is given, one can equip it with the affine connection or the covariant derivative. The further structure of the metric to be preserved by the affine connection, favors the Levi-Civita connection, which is often expressed via the Christoffel symbols, already encountered for the relativistic particle mechanics. This, in turn, defines the Riemann curvature tensor and the Ricci tensor. Numerous additional structures that follow the covariantly constant metric are introduced, such as raising and lowering of indices, Killing vector fields, the volume form, the Hodge star map, geodesics, and geodesic normal coordinates.

The (Riemannian) curvature is based on the notion of a Riemann tensor. Actions in general relativity are found as a generalization of special relativity actions. The action for gravity, the Einstein–Hilbert action, is the simplest nontrivial action compatible with general relativity. Matter is described by the energy-momentum tensor, generalized from special relativity. The equations of motion obtained from the action are the Einstein’s equations.

The metric tensor and the (pseudo-)Riemannian manifolds are defined. The results of the earlier chapters are specialised to this case, in particular the affine connection coefficients are shown to reduce to the Christoffel symbols. The signature of a metric, the timelike, null and spacelike vectors are defined and the notion of a light cone is introduced. It is shown that in two dimensions the notion of curvature agrees with intuition. It is also shown that geodesic lines extremise the interval (i.e. the ‘distance’). Mappings between Riemann spaces are discussed. Conformal curvature (= the Weyl tensor) is defined and it is shown that zero conformal curvature on a manifold of dimension >=4 implies that the metric is proportional to the flat one. Conformal flatness in three dimensions and the Cotton–York tensor are discussed. Embeddings of Riemannian manifolds in Riemannian manifolds of higher dimension are discussed and the Gauss–Codazzi equations derived. The Petrov classification of conformal curvature tensors in four dimensions with signature (+ - - -) is introduced at an elementary level.

With the form of the target theory built up over the previous two chapters, we move to a geometric description of gravitational motion. By recasting the relative dynamics of a pair of falling objects as the deviation of nearby geodesic trajectories in a spacetime with a metric, Einstein’s equation is motivated. To describe geodesic deviation quantitatively, the Riemann tensor is introduced, and its role in characterizing spacetime structure is developed. With the full field equation of general relativity in place, the linearized limit is carefully developed and compared with the gravito-electro-magnetic theory from the first chapter.

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).