Jürgen Jost

Riemannian Geometry and Geometric Analysis (Universitext)

• Buch

Produktdetails

• Verlag: Springer
• ISBN-13: 9783540426271
• ISBN-10: 3540426272
• Artikelnr.: 38072130

Inhaltsangabe

Chapter 1
1. Foundational Material
1.1 Manifolds and Differentiable Manifolds
1.2 Tangent Spaces
1.3 Submanifolds
1.4 Riemannian Metrics
1.5 Vector Bundles
1.6 Integral Curves of Vector Fields. Lie Algebras
1.7 Lie Groups
1.8 Spin Structures Exercises for
Chapter 2
2. De Rham Cohomology and Harmonic Differential Forms
2.1 The Laplace Operator
2.2 Representing cohomology Classes by HarmonicForms
2.3 Generalizations Exercises for
Chapter 3
3. Parallel Transport, Connenctions, and Covariant Derivatives
3.1 Connections in Vector Bundles
3.2 Metric Connections. The Yang-Mills Functional
3.3 The Levi-Civita Connection
3.4 Connections for Spin Structures and the Dirac Operator
3.5 The Bochner Method
3.6 The Geometry of Submanifolds, Minimal Submanifolds Exercises for
Chapter 4
4. Geodesics and Jacobi Fields
4.1 1st and 2nd Variation of Arc Length and Energy
4.2 Jacobi Fields
4.3 Conjugate Points and Distance Minimizing Geodesics
4.4 Riemannian Manifolds of Constant Curvature
4.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates
4.6 Geometric Applications of Jacobi Field Estimates
4.7 Approximate Fundamental Solutions and Representation Formulae
4.8 The geometry of manifolds of nonpositive sectional curvatur Exercises for
Chapter 5 A short Survey on Curvature and Topology
5. Symmetric Spaces and Kähler Manifolds
5.1 Complex Projective Space. Definition of Kähler Manifolds
5.2 The Geometry of Symmetric Spaces
5.3 Some Results about the Structure of Symmetric Saces
5.4 The Space Sl/(n,R)/SO(n,R)
5.5 Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds Exercises for
Chapter 6
6. Morse Theory and Floer Homology
6.1 Preliminaries: Aims of Morse thery
6.2 Compactness: The Palais-Smale condition and the existence of saddle points
6.3 Local analysis: Nondegeneracy of critical points, Morse lemma, stable and unstable manifolds
6.4 Limits of trajectories of the gradient flow
6.5 The Morse-Sma.

Rezensionen

From the reviews of the first and second editions: "... a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. ... The book is made more interesting by the perspectives in various sections, where the author mentions the history and development of the material and provides the reader with references" Mathematical Rezensions " ...the tensorial and intrinsic methods are not separated in the discussions. Instead, the easiest to handle is used for the proofs. ...this symbiosis of these methods gives a deeper understanding for clever readers. ... The book develops the subjects very systematically and takes it beyond the standard introductory topics successfully. I recommend it to everybody, who is already acquainted with Riemannian Geometry, but wants to know it better and deeper or is interested in further nice investigations. Acta Scientarium Mathematicarum.