Symmetric Spaces and Holonomy

Part of the Graduate Texts in Mathematics book series (GTM, volume 171)


In this chapter we give an overview of (locally) symmetric spaces and holonomy. Most standard results are proved or at least mentioned. We give a few 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.


Symmetric Spaces Nonnegative Curvature Operator Bi-invariant Metric Holonomy Group Parallel Curvature Tensor 
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© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

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