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 systematic 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 geometry stimulates progress in geometric analysis by setting ambitious goals.
It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the
methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds.