Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature, … ) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature, … ). By way of contrast, geometric analysis is a perhaps somewhat less system- atic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geom- etry stimulates progress in geometric analysis by setting ambitious goals. It is the aim of this book to be a systematic and comprehensive intro- duction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and an- alytic methods in the study of Riemannian manifolds. The present work is the third edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr-University Bochum and the University of Leipzig. The first main new feature of the third edition is a new chapter on Morse theory and Floer homology that attempts to explain the relevant ideas and concepts in an elementary manner and with detailed examples.
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