Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface to the Second Edition -- Preface -- I Riemannian Manifolds -- I.1. Connections -- I.2. Parallel Translation of Vector Fields -- I.3. Geodesics and the Exponential Map -- I.4. The Torsion and Curvature Tensors -- I.5. Riemannian Metrics -- I.6. The Metric Space Structure -- I.7. Geodesics and Completeness -- I.8. Calculations with Moving Frames -- I.9. Notes and Exercises -- Bibliographic Sampler -- Lie Brackets of Vector Fields -- Connections and Covariant Differentiation -- The Second Bianchi Identity -- Covariant Differentiation in Vector Bundles -- Coordinate Characterization of T = 0 -- Gradients -- On Theorem I.6.1 -- The Hopf-Rinow Theorems -- Length Spaces -- Moving Frames -- Hessians -- Moving Frames in Euclidean Space -- Examples -- II Riemannian Curvature -- II.1. The Riemann Sectional Curvature -- II.2. Riemannian Submanifolds -- The Second Fundamental Form Via Moving Frames -- II.3. Spaces of Constant Sectional Curvature -- Euclidean Space -- Spheres: The First Method -- Spheres: The Second Method -- Some Generalities about Isometries -- Spheres: The Third Method -- Hyperbolic Space -- II.4. First and Second Variations of Arc Length -- II.5. Jacobi's Equation and Criteria -- II.6. Elementary Comparison Theorems -- II.7. Jacobi Fields and the Exponential Map -- II.8. Riemann Normal Coordinates -- Normal Coordinates in Constant Sectional Curvature Spaces -- II.9. Notes and Exercises -- Curvature Tensor Estimates -- Schur's Theorem -- Bibliographic Sampler for Curves and Surfaces in Euclidean Space -- Totally Geodesic Submanifolds -- Two Norms of Linear Transformations -- The Second Fundamental Form and Local Convexity -- Mean Curvature -- Hessians, Again -- Connections in Normal Bundles of Submanifolds -- On Isometries.

Spherical and Hyperbolic Geometry -- A Result of J. L. Synge -- On Conjugate Points -- On Jacobi's Criteria -- Geometry of the Index Form -- On Manifolds of Positive Curvature -- On the Morse-Schönberg Theorem -- On the Rauch Theorem -- Riemann Normal Coordinates -- III Riemannian Volume -- III.1. Geodesic Spherical Coordinates -- III.2. The Conjugate and Cut Loci -- III.3. Riemannian Measure -- The Effective Calculation of Integrals -- Volume of Metric Disks -- III.4. Volume Comparison Theorems -- III.5. The Area of Spheres -- III.6. Fermi Coordinates -- III.7. Integration of Differential Forms -- Green's Formulae in Riemannian Manifolds -- III.8. Notes and Exercises -- Riemannian Symmetric Spaces -- The Cut Locus -- Two-Point Homogeneous Spaces -- On the Riemannian Measure -- The Area Formula -- The Smooth Coarea Formula -- The First Variation of Area -- Hypersurfaces of Constant Mean Curvature -- The Gauss Map -- On the Günther-Bishop Theorems -- On Volumes of Disks -- Fermi Coordinates -- H. Weyl's Formula for the Volume of Tubes -- Geometric Interpretation of the Riccati Equation -- Manifolds With No Conjugate Points -- The Laplacian -- The Second Variation of Area -- III.9. Appendix: Eigenvalue Comparison Theorems -- Notes and Exercises -- Max-Min Methods -- Weyl's Asymptotic Formula -- Eigenvalues and Wirtinger's Inequality -- When the Fundamental Tone Is Bounded Away From 0 -- Lichnerowicz's Formula -- IV Riemannian Coverings -- IV.1. Riemannian Coverings -- IV.2. The Fundamental Group -- IV.3. Volume Growth of Riemannian Coverings -- IV.4. Discretization of Riemannian Manifolds -- IV.5. The Free Homotopy Classes -- IV.6. Notes and Exercises -- Parallel Translation and Curvature -- The Myers-Steenrod Theorem -- Deck Transformation Groups, Discrete Groups, and Tori -- Global Cartan-Ambrose-Hicks Theorem -- On the Myers Comparison Theorem.

Manifolds of Nonpositive Curvature -- Fundamental Domains -- Coverings by Compacta -- On Theorem IV.4.1 -- Homotopy Considerations -- The Results in Length Spaces -- On the Displacement Norm -- Existence of Closed Geodesics -- V Surfaces -- V.1. Systolic Inequalities -- V.2. Gauss-Bonnet Theory of Surfaces -- The Umlaufsatz -- Applications of the Umlaufsatz -- V.3. The Collar Theorem -- V.4. The Isoperimetric Problem: Introduction -- V.5. Surfaces with Curvature Bounded from Above -- The Theorems of Carleman and Weil -- The Bol-Fiala Inequalities -- V.6. The Isoperimetric Problem on the Paraboloid of Revolution -- The Curvature Flow: Background -- The Isoperimetric Inequality -- The Paraboloid of Revolution -- V.7. Notes and Exercises -- Systolic Inequalities -- The Gauss-Bonnet Theorem -- On Randol's Collar Theorem -- The Theorems of Carleman and Weil -- Curvature Flow in Higher Dimensions -- Further Isoperimetric Inequalities on Surfaces -- VI Isoperimetric Inequalities (Constant Curvature) -- VI.1. The Brunn-Minkowski Theorem -- VI.2. Solvability of a Neumann Problem in Rn -- VI.3. Fermi Coordinates in Constant Sectional Curvature Spaces -- VI.4. Spherical Symmetrization and Isoperimetric Inequalities -- The Argument in Hyperbolic Space -- The Argument in the Sphere -- VI.5. M. Gromov's Uniqueness Proof - Euclidean and Hyperbolic Space -- VI.6. The Isoperimetric Inequality on Spheres -- VI.7. Notes and Exercises -- Symmetrization -- Other Methods -- The Elementary Version of Steiner Symmetrization -- The Faber-Krahn Inequality -- Other Developments -- VII The Kinematic Density -- VII.1. The Differential Geometry of Analytical Dynamics -- The Canonical Symplectic Form on Cotangent Bundles -- Lagrangian-Induced Symplectic Forms on Tangent Bundles -- Newton's Equations in Riemannian Geometry -- The Liouville Theorems.

VII.2. The Berger-Kazdan Inequalities -- The Blaschke Conjecture -- VII.3. On Manifolds with No Conjugate Points -- VII.4. Santalo's Formula -- VII.5. Notes and Exercises -- The Kinematic Density Via Moving Frames -- The Differential of the Geodesic Flow -- Manifolds Without Conjugate Points -- A Variant of E. Hopf's Theorem -- The Osserman-Sarnak Inequality -- Aufwiedersehnsfläche -- An Eigenvalue Inequality -- VIII Isoperimetric Inequalities (Variable Curvature) -- VIII.1. Croke's Isoperimetric Inequality -- VIII.2. Buser's Isoperimetric Inequality -- VIII.3. Isoperimetric Constants -- Modified Isoperimetric Constants -- VIII.4. Discretizations and Isoperimetry -- VIII.5. Notes and Exercises -- Analytic Isoperimetric Inequalities -- The Compact Case -- Buser's Isoperimetric Inequality -- IX Comparison and Finiteness Theorems -- IX.1. Preliminaries -- IX.2. H. E. Rauch's Comparison Theorem -- IX.3. Comparison Theorems with Initial Submanifolds -- IX.4. Refinements of the Rauch Theorem -- IX.5. Triangle Comparison Theorems -- IX.6. Convexity -- Calculations Associated with Convexity -- Applications -- IX.7. Center of Mass -- IX.8. Cheeger's Finiteness Theorem -- IX.9. Notes and Exercises -- The Rauch Comparison and Sphere Theorems -- Diameter Sphere Theorems -- The Alexandrov-Toponogov Comparison Theorems -- Busemann Functions and Halfspaces -- Finiteness Theorems -- Convergence Theorems for Riemannian Manifolds -- Hints and Sketches for Exercises -- Hints and Sketches: Chapter I -- Hints and Sketches: Chapter II -- Hints and Sketches: Chapter III -- Hints and Sketches: Chapter IV -- Hints and Sketches: Chapter V -- Hints and Sketches: Chapter VI -- Hints and Sketches: Chapter VII -- Hints and Sketches: Chapter VIII -- Hints and Sketches: Chapter IX -- Bibliography -- Author Index -- Subject Index.