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                                         Holonomy groups in riemannian geometry
                                                    Andrew Clarke and Bianca Santoro 1
                                                                June 15, 2012
                                           1The first author would like to acknowledge the support of FAPESP
                                        (Processo 2011/07363-6) in the preparation of these notes. The second
                                        author acknowledges the support of the PSC-CUNY (award number 64059-
                                        00 42) and of the National Science Foundation under grant DMS 1007155.
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                                          Preface
                     These are the very unpretentious lecture notes for the minicourse
                   HolonomyGroupsinRiemanniangeometry,apartoftheXVIIBrazil-
                   ian School of Geometry, to be held at UFAM (Amazonas, Brazil), in
                   July of 2012. We have aimed at providing a first introduction to
                   the main general ideas on the study of the holonomy groups of a
                   Riemannian manifold.
                     The holonomy group is one of the fundamental analytical objects
                   that one can define on a riemannian manfold. It interacts with, and
                   contains information about a great number of geometric properties
                   of the manifold. Namely, the holonomy group detects the local re-
                   ducibility of the manifold, and also whether the metric is locally
                   symmetric.
                     Wehavestructured these notes to emphasize the principal bundle
                   formulation. This is done because it was felt that the symmetries of
                   a manifold, even pointwise, are best expressed in this way. We have
                   striven though to give constructions in two different ways, on vector
                   bundles and principal bundles, and also to stress the ways in which
                   these perspectives relate.
                     We plan to keep improving these notes, and the updates will be
                   available at
                     http://www.sci.ccny.cuny.edu/∼bsantoro/ and
                     http://www.ime.usp.br/∼clarke/en/
                     Meanwhile, we invite the reader to send suggestions, comments
                   and possible corrections to
                     bsantoro@ccny.cuny.edu or clarke@ime.usp.br
                     Disclaimer: Any opinions, findings, and conclusions or recommen-
                   dations expressed in this material are those of the authors and do not
                   necessarily reflect the views of the National Science Foundation.
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