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As Sasakian manifolds are all examples of Riemannian foliations with one-dimensional leaves, this chapter goes into the world of foliations with a particular focus on the Riemannian case. Haeiger structures, leaf holonomy, the holonomy groupoid, and basic cohomology are introduced, focusing on the transverse geometry of Riemannian foliations, in particular, Riemannian ows. Certain invariants such as the toral rank are introduced.

This chapter begins by introducing various geometries that play important roles in the way they relate to Sasakian structures. It espouses the point of view that a geometric structure is best described as a G-structure which may or may not be (partially) integrable. Some selected topics include: Riemannian metrics, complex structures, symplectic structures, contact structures, quaternionic structures, group actions, pseudogroups, sheaves, bundles, connections, holonomy, curvature and integrability.

This chapter gives a very brief overview of the rich theory of Killing spinors describing Bär's correspondence which connects Sasaki-Einstein geometry to Berger's holonomy theorem. It is then shown how weak holonomy G2 and nearly Kähler structures relate to 3-Sasakian 7-manifolds and Sasaki-Einstein 5-manifolds. The chapter ends with a brief discussion of how Sasakian geometry naturally appears in various supersymmetric physical theories such as supergravity theory, superstring theory, and M-theory. In particular, Sasaki-Einstein 5-manifolds play a crucial role in the AdS/CFT duality conjecture.

In the early analyses of metrics with special holonomy in dimensions 7 and 8, particularly in regards to their existence and generality, heavy use was made of the Cartan–Kähler theorem, essentially because the analyses were reduced to the study of overdetermined PDE systems whose natures were complicated by their diffeomorphism invariance. The Cartan–Kähler theory is well suited for the study of such systems and the local properties of their solutions. However, the Cartan–Kähler theory is not particularly well suited for studies of global problems for two reasons: first, it is an approach to PDE that relies entirely on the local solvability of initial value problems and, second, the Cartan–Kähler theory is only applicable in the real-analytic category. Nevertheless, when there are no other adequate methods for analyzing the local generality of such systems, the Cartan–Kähler theory is a useful tool and it has the effect of focusing attention on the initial value problem as an interesting problem in its own right. This chapter clarifies some of the existence issues involved in applying the initial value problem to the problem of constructing metrics with special holonomy. In particular, it discusses the role of the assumption of real-analyticity and presents examples to show that one cannot generally avoid such assumptions in the initial value formulations of these problems.

This chapter introduces Yang-Mills theories as a natural generalization of Maxwell's theory. It discusses covariant derivatives and the Yang-Mills equations, which it also considers in a Hamiltonian fashion, and presents the notion of holonomy as a generalization of the notion of circulation of a vector. The chapter also discusses the path-ordered exponential, the non-Abelian Stokes theorem, and Giles' theorem.