Preface XI There are some glaring omissions: The Laplace-Beltrami operator is treated only summarily, there is no spectral theory, and the structure theory of Lie algebras is not treated and used. Thus the finer theory of symmetric spaces is outside of the scope of this book. The exposition is not always linear. Sometimes concepts treated in detail in later sections are used or pointed out earlier on when they appear in a natural way. Text cross-references to sections, subsections, theorems, numbered equations, items in a list, etc., appear in parantheses, for example, section (1), subsection (1.1), theorem (3.16), equation (3.16.3) which will be called (3) within (3.16) and its proof, property (3.22.1). This book grew out of lectures which I have given during the last three decades on advanced differential geometry, Lie groups and their actions, Riemann geometry, and symplectic geometry. I have benefited a lot from the advise of colleagues and remarks by readers and students. In particular I want to thank Konstanze Rietsch whose write-up of my lecture course on isometric group actions was very helpful in the preparation of this book and Simon Hochgerner who helped with the last section. Support by the Austrian FWF-projects P 4661, P 7724-PHY, P 10037- MAT, P 14195-MAT, and P 17108-N04 during the years 1983 - 2007 is acknowledged. Kritzendorf, April 2008

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