Symmetric Spaces and Holonomy
In this chapter we shall give a brief overview of (locally) symmetric spaces and holonomy. Only the simplest proofs will be presented. Thus, we will have to be sketchy in places. Still, most of the standard results are proved or at least mentioned. We give some explicit examples, including the complex projective space, in order to show how one can compute curvatures on symmetric spaces relatively easily. There is a brief introduction to holonomy and the de Rham decomposition theorem. We give a few interesting consequences of this theorem and then proceed to discuss how holonomy and symmetric spaces are related Finally, we classify all compact manifolds with nonnegative curvature operator. We shall in a few places use results from Chapter 9. They will therefore have to be taken for granted at this point.
KeywordsRiemannian Manifold Symmetric Space Sectional Curvature Curvature Tensor Curvature Operator
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