Preface
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 fourth 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. Besides several smaller additions, reorganizations, corrections (I
am grateful to J.Weber and P.Hinow for useful comments), and a systematic
bibliography, the main new features of the present edition are a systematic in-
troduction to K¨ahler geometry and the presentation of additional techniques
from geometric analysis.
Let me now briefly describe the contents:
In the first chapter, we introduce the basic geometric concepts, like dif-
ferentiable manifolds, tangent spaces, vector bundles, vector fields and one-
parameter groups of diffeomorphisms, Lie algebras and groups and in par-
ticular Riemannian metrics. We also derive some elementary results about
geodesics.
The second chapter introduces de Rham cohomology groups and the es-
sential tools from elliptic PDE for treating these groups. In later chapters,
we shall encounter nonlinear versions of the methods presented here.
The third chapter treats the general theory of connections and curvature.
In the fourth chapter, we introduce Jacobi fields, prove the Rauch com-
parison theorems for Jacobi fields and apply these results to geodesics.
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