Riemannian Geometry

From the preface:Many years have passed since the first edition. However, the encouragements of various readers and friends have persuaded us to write this third edition. During these years, Riemannian Geometry has undergone many dramatic developments. Here is not the place to relate them. The reader can consult for instance the recent book [Br5]. of our mentor Marcel Berger. However, Riemannian Geometry is not only a fascinating field in itself. It has proved to be a precious tool in other parts of mathematics. In this respect, we can quote the major breakthroughs in four-dimensional topology which occurred in the eighties and the nineties of the last century (see for instance [L2]). These have been followed, quite recently, by a possibly successful approach to the Poincaré conjecture. In another direction, Geometric Group Theory, a very active field nowadays (cf. [Gr6]), borrows many ideas from Riemannian or metric geometry. But let us stop hogging the limelight. This is justa textbook. We hope that our point of view of working intrinsically with manifolds as early as possible, and testing every new notion on a series of recurrent examples (see the introduction to the first edition for a detailed description), can be useful both to beginners and to mathematicians from other fields, wanting to acquire some feeling for the subject.

Includes supplementary material: sn.pub/extras

Klappentext

From the preface:Many years have passed since the first edition. However, the encouragements of various readers and friends have persuaded us to write this third edition. During these years, Riemannian Geometry has undergone many dramatic developments. Here is not the place to relate them. The reader can consult for instance the recent book [Br5]. of our mentor Marcel Berger. However, Riemannian Geometry is not only a fascinating field in itself. It has proved to be a precious tool in other parts of mathematics. In this respect, we can quote the major breakthroughs in four-dimensional topology which occurred in the eighties and the nineties of the last century (see for instance [L2]). These have been followed, quite recently, by a possibly successful approach to the Poincaré conjecture. In another direction, Geometric Group Theory, a very active field nowadays (cf. [Gr6]), borrows many ideas from Riemannian or metric geometry. But let us stop hogging the limelight. This is justa textbook. We hope that our point of view of working intrinsically with manifolds as early as possible, and testing every new notion on a series of recurrent examples (see the introduction to the first edition for a detailed description), can be useful both to beginners and to mathematicians from other fields, wanting to acquire some feeling for the subject.

Inhalt
1 Differential manifolds.- 1.A From submanifolds to abstract manifolds.- 1.B The tangent bundle.- 1.C Vector fields.- 1.D Baby Lie groups.- 1.E Covering maps and fibrations.- 1.F Tensors.- 1.G. Differential forms.- 1.H Partitions of unity.- 2 Riemannian metrics.- 2.A Existence theorems and first examples.- 2.B Covariant derivative.- 2.C Geodesies.- 2.D A glance at pseudo-Riemannian manifolds.- 3 Curvature.- 3.A. The curvature tensor.- 3.B. First and second variation.- 3.C. Jacobi vector fields.- 3.D. Riemannian submersions and curvature.- 3.E. The behavior of length and energy in the neighborhood of a geodesic.- 3.F Manifolds with constant sectional curvature.- 3.G Topology and curvature: two basic results.- 3.H. Curvature and volume.- 3.I. Curvature and growth of the fundamental group.- 3.J. Curvature and topology: some important results.- 3.K. Curvature tensors and representations of the orthogonal group.- 3.L. Hyperbolic geometry.- 3.M. Conformai geometry.- 4 Analysis on manifolds.-4.A. Manifolds with boundary.- 4.B. Bishop inequality.- 4.C. Differential forms and cohomology.- 4.D. Basic spectral geometry.- 4.E. Some examples of spectra.- 4.F The minimax principle.- 4.G Eigenvalues estimates.- 4.H. Paul Levy's isoperimetric inequality.- 5 Riemannian submanifolds.- 5.A. Curvature of submanifolds.- 5.B Curvature and convexity.- 5.C Minimal surfaces.- A Some extra problems.- B Solutions of exercises.- List of figures.