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Nigel Hitchin has played a key role in the exploration of four-dimensional Riemannian geometry, and in particular has made foundational contributions to the theory of self-dual manifolds, four-dimensional Einstein manifolds, spinc structures, and Kähler geometry. In the process, he called attention to the profound mathematical interest of beautiful geometric problems that had previously only been considered by physicists. This chapter focuses on an interesting relationship between the four-dimensional Einstein–Maxwell equations and Kähler geometry, and points out some fascinating open problems that directly impinge on this relationship.

This chapter focuses on gravitation. It first reviews some basic concepts of the Riemannian geometry and establishes notation. It then discusses the gravitational action, specifically the fermionic action. It introduces Einstein-, Lorentz-, and Weyl anomalies by violating the corresponding Einstein-, Lorentz-, and Weyl symmetries, and establishes consistency conditions. The equivalence of the Einstein- and Lorentz anomaly is demonstrated, and the covariant anomaly is discussed. Finally, the chapter treats gravitation on a BRS level, deriving the SZ chain of descent equations. Index theorems are use to carry out explicit anomaly examples.

The key notions of the differential and Riemannian geometry necessary for understanding the General Relativity are introduced here. We provide the reader with the necessary tools for study the properties of black holesand their interaction with matter and fields. We tried to omit unnecessary details and to make the presentation simpler. We focus on an explanation of the basic concepts and explicit formulas, which can be used for concrete calculations, and give hints, which might be used to simplify these calculations.

This chapter deals with the foundations of geometry, with emphasis on Helmholtz’s approach to this problem in the late 1860s. It is first recalled how the ancient belief in the apodictic truth of the axioms of geometry came under attack at the turn of the eighteenth and nineteenth centuries, making room for the new geometries of Gauss, Bolyai, Lobachevski, and Riemann. This opening of possibilities raised the question of the necessary structural kernel of any physical geometry. The central section of this chapter is a detailed account of Helmholtz’s answer to this question, according to which the locally Euclidean character of space follows from its measurability by freely mobile rigid bodies. This argument has difficulties, due to a seeming circularity in the definition of rigid bodies and to the strictness of the assumed rigidity, which requires spaces of constant curvature. It is shown how these difficulties can be circumvented, leading to the necessity of the Riemannian structure of space (with any curvature) for measurable space.

Riemannian geometry deals with distances of infinitesimally close points. A curve of minimal length is a geodesic. This variational principle happens to have the same structure as that of mechanics, therefore mechanical methods can be applied. Isometries of metrics lead to conservation laws. Conversely, the path of a Newtonian dynamical system can be reformulated as a geodesic of a metric in configuration space: the Maupertuis principle. Yet another application is to Fermat's principle of optics: the refractive index determines a metric, whose geodesics are light rays. But the deeper application of Riemannian geometry is to General Relativity: the gravitational field is the metric tensor. This chapter determines the orbits of a particle in the field of a black hole.

The Euler equations of a rigid body can be understood as the geodesic equations for a metric on the rotation group. A rapid introduction to the Riemannian geometry of Lie groups (following Milnor) is given and illuminated by the example of the rigid body. The deep generalization of Arnold to the case of an incompressible fluid is then explained. The Euler equations of an ideal incompressible fluid are shown to be geodesics of the group of volume preserving diffeomorphisms. The curvature of this metric is calculated. Contrary to the case of the rigid body, the curvature is negative, implying that the dynamics of such a fluid is highly unstable. Some ideas on how geodesic dynamics is modified by dissipation are introduced. This leads to new generalizations of Riemannian geometry.

In general relativity (GR) physics laws are not changed under general coordinate transformations. Among Einstein’s motivation for GR was his wish to have a deeper understanding of the empirically observed equality between gravitational and inertial masses. Gravity naturally enters into the theory when accelerated frames are considered. The generalization of this equivalence principle to electromagnetism implies gravitational redshift, gravitational time dilation, and gravitational bending of a light ray. Most importantly, such considerations led Einstein to the idea that the gravitational effect on a body can be attributed directly to some underlying spacetime feature and the gravitational field is simply curved spacetime. We present some elements of Riemannian geometry (the mathematics of curved space): Gaussian coordinates metric tensor, the geodesic equation, and curvature.

A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors. This chapter proves an analogous jet isomorphism theorem for conformal geometry. By making conformal changes, the Taylor expansion of a metric in geodesic normal coordinates can be further simplified, resulting in a “conformal normal form” for metrics about a point.

The mathematics of a curved manifold is Riemannian geometry. This chapter first presents some basic elements of Gaussian coordinates and the metric tensor. From the requirement that a curve have the shortest length, the calculus of variations is used to deduce the geodesic equation. A geometric description of the equivalence-principle-based physics of gravitational time dilation is presented. Einstein’s motivation for considering curved spacetime as a gravitational field is then discussed. In this geometric theory, the metric plays the role of the relativistic gravitational potential. The interpretation of curved spacetime as a gravitational field naturally suggests that the geodesic equation in spacetime is the general relativity equation of motion, which reduces to the Newtonian equation of motion in the limit of a nonrelativistic moving particle in a static and weak gravitational field. The gravitational redshift effect is shown to follow directly from a curved spacetime (with curvature in the time direction).

The aim of this chapter is to set the ground for the remainder of this work. We present our conventions and notations in Section 1.1. We review some basic facts about the topology of manifolds in Section 1.2. Lorentzian manifolds and spacetimes are introduced in Section 1.3. Elementary facts concerning the Levi-Civita connection and its curvature are reviewed in Section 1.4. Some properties of geodesics, as needed in the remainder of this book, are presented in Section 1.5. The formalism of moving frames is outlined in Section 1.6, where it is used to calculate the curvature of metrics of interest.