Linear Connections and Riemannian Geometry
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We present the general theory of linear connections together with the reduction theory of the frame bundle and a discussion of the corresponding compatible connections. Such reductions are known as H-structures and lead to a unified view on all the geometric structures a manifold may be endowed with: almost complex, pseudo-Riemannian, conformal, almost Hermitean and almost symplectic structures. Further, we discuss the relation between curvature and holonomy and give an introduction to the theory of symmetric spaces. Finally, we present elements of Hodge theory and discuss special properties of four-dimensional Riemannian manifolds.