Holdings: A panoramic view of Riemannian geometry

A panoramic view of Riemannian geometry:

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Bibliographic Details
Main Author:	Berger, Marcel 1927-2016 (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2003
Subjects:	Riemannsche Geometrie Geometry, Riemannian
Online Access:	Inhaltsverzeichnis
Physical Description:	XXIII, 824 S. Ill., graph. Darst.
ISBN:	3540653171 9783642621215

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MARC


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adam_text	Contents Euclidean Geometry ....................................... 1 l.ł Preliminaries .......................................... 2 1.2 Distance Geometry ..................................... 2 1.2.1 A Basic Formula ................................ 2 1.2.2 The Length of a Path ........................... 3 1.2.3 The First Variation Formula and Application to Billiards ...................... 4 1.3 Plane Curves .......................................... 9 1.3.1 Length ........................................ 9 1.3.2 Curvature ..................................... 12 1.4 Global Theory of Closed Plane Curves .................... 18 1.4.1 Obvious Truths About Curves Which are Hard to Prove ........................ 18 1.4.2 The Four Vertex Theorem ....................... 20 1.4.3 Convexity with Respect to Arc Length ............ 22 1.4.4 Umlaufsatz with Corners ........................ 23 1.4.5 Heat Shrinking of Plane Curves ................... 24 і .4.6 Arnol d s Revolution in Plane Curve Theory ....... 24 1.5 The Isoperimetric Inequality for Curves ................... 26 1.6 The Geometry of Surfaces Before and After Gauß.......... 29 1.6.1 Inner Geometry: a First Attempt ................. 30 1.6.2 Looking for Shortest Curves: Geodesies ............ 33 1.6.3 The Second Fundamental Form and Principal Curvatures ........................ 45 1.6.4 The Meaning of the Sign of К .................... 52 1.6.5 Global Surface Geometry ........................ 55 1.6.6 Minimal Surfaces ............................... 58 1.6.7 The Hartman-Nirenberg Theorem for Inner Flat Surfaces .......................... 62 1.6.8 The Isoperimetric Inequality in E3 à la Gromov ___ 63 1.6.8.1 Notes .................................. 66 1.7 Generic Surfaces ....................................... 66 1.8 Heat and Wave Analysis in E2 ........................... 70 1.8.1 Planar Physics ................................. 70 1.8.1.1 Bibliographical Note ..................... 71 XII Contents 1.8.2 Why the Eigenvalue Problem? .................... 71 1.8.3 Minimax ...................................... 75 1.8.4 Shape of a Drum ............................... 78 1.8.4.1 A Few Direct Problems .................. 79 1.8.4.2 The Faber-Krahn Inequality .............. 81 1.8.4.3 Inverse Problems ........................ 83 1.8.5 Heat .......................................... 87 1.8.5.1 Eigenfunctions .......................... 90 1.8.6 Relations Between the Two Spectra ............... 91 1.9 Heat and Waves in E3,Ed and on the Sphere .............. 94 1.9.1 Euclidean Spaces ............................... 94 1.9.2 Spheres ........................................ 95 1.9.3 Billiards in Higher Dimensions ................... 97 1.9.4 The Wave Equation Versus the Heat Equation ...... 98 2 Transition ..................................................101 3 Surfaces from Gauß to Today ..............................105 3.1 Gauß.................................................105 3.1.1 Theorema Egregium ............................105 3.1.1.1 The First Proof of Gauß s Theorema Egregium; the Concept of ds2 .............106 3.1.1.2 Second Proof of the Theorema Egregium ... 109 3.1.2 The Gauß-Bonnet Formula and the Rodrigues-Gauß Map.................... Ill 3.1.3 Parallel Transport ..............................113 3.1.4 Inner Geometry ................................116 3.2 Alexandrov s Theorems .................................120 3.2.1 Angle Corrections of Legendre and Gauß in Geodesy .....................................123 3.3 Cut Loci ..............................................125 3.4 Global Surface Theory ..................................131 3.4.1 Bending Surfaces ...............................131 3.4.1.1 Bending Polyhedra ......................132 3.4.1.2 Bending and Wrinkling with Little Smoothness ..................133 3.4.2 Mean Curvature Rigidity of the Sphere ............134 3.4.3 Negatively Curved Surfaces ......................135 3.4.4 The Willmore Conjecture ........................136 3.4.5 The Global Gauß-Bonnet Theorem for Surfaces ___136 3.4.6 The Hopf Index Formula .........................139 Contents XIII Riemann s Blueprints ......................................143 4.1 Smooth Manifolds .....................................143 4.1.1 Introduction ...................................143 4.1.2 The Need for Abstract Manifolds .................146 4.1.3 Examples ......................................149 4.1.3.1 Submanifolds ...........................151 4.1.3.2 Products ...............................151 4.1.3.3 Lie Groups .............................152 4.1.3.4 Homogeneous Spaces ....................152 4.1.3.5 Grassmannians over Various Algebras ......153 4.1.3.6 Gluing .................................156 4.1.4 The Classification of Manifolds ...................157 4.1.4.1 Surfaces ...............................158 4.1.4.2 Higher Dimensions ......................159 4.1.4.3 Embedding Manifolds in Euclidean Space .. 161 4.2 Calculus on Manifolds ..................................162 4.2.1 Tangent Spaces and the Tangent Bundle ...........162 4.2.2 Differential Forms and Exterior Calculus ...........166 4.3 Examples of Riemann s Definition ........................172 4.3.1 Riemann s Definition ............................172 4.3.2 Hyperbolic Geometry ...........................176 4.3.3 Products, Coverings and Quotients ................183 4.3.3.1 Products ...............................183 4.3.3.2 Coverings ..............................184 4.3.4 Homogeneous Spaces ............................186 4.3.5 Symmetric Spaces ..............................189 4.3.5.1 Classification ...........................192 4.3.5.2 Rank ..................................193 4.3.6 Riemannian Submersions ........................194 4.3.7 Gluing and Surgery .............................196 4.3.7.1 Gluing of Hyperbolic Surfaces ............196 4.3.7.2 Higher Dimensional Gluing ...............198 4.3.8 Classical Mechanics .............................199 4.4 The Riemann Curvature Tensor .........................200 4.4.1 Discovery and Definition .........................200 4.4.2 The Sectional Curvature .........................204 4.4.3 Standard Examples .............................207 4.4.3.1 Constant Sectional Curvature .............207 4.4.3.2 Projective Spaces KP ...................209 4.4.3.3 Products ...............................209 4.4.3.4 Homogeneous Spaces ....................210 4.4.3.5 Hypersurfaces in Euclidean Space .........211 XIV Contents 4.5 A Naive Question: Does the Curvature Determine the Metric? ..................................213 4.5.1 Surfaces .......................................214 4.5.2 Any Dimension .................................215 4.6 Abstract Riemannian Manifolds .........................216 4.6.1 Isometrically Embedding Surfaces in E3 ...........217 4.6.2 Local Isometric Embedding of Surfaces in E3 .......217 4.6.3 Isometric Embedding in Higher Dimensions ........218 5 A One Page Panorama .....................................219 6 Metric Geometry and Curvature ...........................221 6.1 First Metric Properties .................................222 6.1.1 Local Properties ................................222 6.1.2 Hopf-Rinow and de Rham Theorems ..............226 6.1.2.1 Products ...............................229 6.1.3 Convexity and Small Balls .......................229 6.1.4 Totally Geodesic Submanifolds ...................231 6.1.5 Center of Mass .................................233 6.1.6 Examples of Geodesies ..........................235 6.1.7 Transition .....................................238 6.2 First Technical Tools ...................................239 6.3 Second Technical Tools .................................248 6.3.1 Exponential Map ...............................248 6.3.1.1 Rank ..................................250 6.3.2 Space Forms ...................................251 6.3.3 Nonpositive Curvature ..........................254 6.4 Triangle Comparison Theorems ..........................257 6.4.1 Bounded Sectional Curvature ....................257 6.4.2 Ricci Lower Bound ..............................262 6.4.3 Philosophy Behind These Bounds .................267 6.5 Injectivity, Convexity Radius and Cut Locus ..............268 6.5.1 Definition of Cut Points and Injectivity Radius .....268 6.5.2 Klingenberg and Cheeger Theorems ...............272 6.5.3 Convexity Radius ...............................278 6.5.4 Cut Locus .....................................278 6.5.5 Blaschke Manifolds .............................285 6.6 Geometric Hierarchy ...................................286 6.6.1 The Geometric Hierarchy ........................289 6.6.1.1 Space Forms ............................289 6.6.1.2 Rank 1 Symmetric Spaces ................289 6.6.1.3 Measure Isotropy ........................289 6.6.1.4 Symmetric Spaces .......................290 6.6.1.5 Homogeneous Spaces ....................290 Contents XV 6.6.2 Constant Sectional Curvature ....................290 6.6.2.1 Negatively Curved Space Forms in Three and Higher Dimensions ..........292 6.6.2.2 Mostów Rigidity ........................293 6.6.2.3 Classification of Arithmetic and Nonarithmetic Negatively Curved Space Forms ............................294 6.6.2.4 Volumes of Negatively Curved Space Forms 295 6.6.3 Rank 1 Symmetric Spaces .......................295 6.6.4 Higher Rank Symmetric Spaces ...................296 6.6.4.1 Superrigidity ...........................296 6.6.5 Homogeneous Spaces ............................296 Volumes and Inequalities on Volumes of Cycles ............299 7.1 Curvature Inequalities ..................................299 7.1.1 Bounds on Volume Elements and First Applications . 299 7.1.1.1 The Canonical Measure ..................299 7.1.1.2 Volumes of Standard Spaces ..............303 7.1.1.3 The Isoperimetric Inequality for Spheres ... 304 7.1.1.4 Sectional Curvature Upper Bounds ........305 7.1.1.5 Ricci Curvature Lower Bounds ............308 7.1.2 Isoperimetric Profile ............................315 7.1.2.1 Definition and Examples .................315 7.1.2.2 The Gromov-Bérard-Besson-Gallot Bound . 319 7.1.2.3 Nonpositive Curvature on Noncompact Manifolds ................322 7.2 Curvature Free Inequalities on Volumes of Cycles ..........325 7.2.1 Curves in Surfaces ..............................325 7.2.1.1 Loewner, Pu and Blatter-Bavard Theorems 325 7.2.1.2 Higher Genus Surfaces ...................329 7.2.1.3 The Sphere .............................336 7.2.1.4 Homological Systoles ....................338 7.2.2 Inequalities for Curves ...........................340 7.2.2.1 The Problem, and Standard Manifolds .....340 7.2.2.2 Filling Volume and Filling Radius .........342 7.2.2.3 Gromov s Theorem and Sketch of the Proof 344 7.2.3 Higher Dimensional Systoles: Systolic Freedom Almost Everywhere ..............348 7.2.4 Embolie Inequalities ............................353 7.2.4.1 Introduction ............................353 7.2.4.2 The Unit Tangent Bundle ................357 7.2.4.3 The Core of the Proof ...................359 7.2.4.4 Crofce s Three Results ...................363 7.2.4.5 Infinite Injectivity Radius ................366 7.2.4.6 Using Embolie Inequalities ...............367 XVI Contents 8 Transition: The Next Two Chapters ........................369 8.1 Spectral Geometry and Geodesic Dynamics ...............369 8.2 Why are Riemannian Manifolds So Important? ............372 8.3 Positive Versus Negative Curvature ......................372 9 Spectrum of the Laplacian .................................373 9.1 History ...............................................374 9.2 Motivation ............................................375 9.3 Setting Up ............................................376 9.3.1 Xdefinition .....................................376 9.3.2 The Hodge Star ................................378 9.3.3 Facts ..........................................380 9.3.4 Heat, Wave and Schrödinger Equations ............381 9.4 Minimax ..............................................383 9.4.1 The Principle ..............................___383 9.4.2 An Application .................................385 9.5 Some Extreme Examples ................................387 9.5.1 Square Tori, Alias Several Variable Fourier Series ... 387 9.5.2 Other Flat Tori .................................388 9.5.3 Spheres ........................................390 9.5.4 ЮГ ..........................................390 9.5.5 Other Space Forms .............................391 9.6 Current Questions .....................................392 9.6.1 Direct Questions About the Spectrum .............392 9.6.2 Direct Problems About the Eigenfunctions .........393 9.6.3 Inverse Problems on the Spectrum ................393 9.7 First Tools: The Heat Kernel and Heat Equation ...........393 9.7.1 The Main Result ...............................393 9.7.2 Great Hopes ...................................396 9.7.3 The Heat Kernel and Ricci Curvature .............401 9.8 The Wave Equation: The Gaps ..........................402 9.9 The Wave Equation: Spectrum & Geodesic Flow ...........405 9.10 The First Eigenvalue ...................................408 9.10.1 λι and Ricci Curvature ..........................408 9.10.2 Cheeger s Constant .............................409 9.10.3 λι and Volume; Surfaces and Multiplicity ..........410 9.10.4 Kahler Manifolds ...............................411 9.11 Results on Eigenfunctions ...............................412 9.11.1 Distribution of the Eigenfunctions ................412 9.11.2 Volume of the Nodal Hypersurfaces ...............413 9.11.3 Distribution of the Nodal Hypersurfaces ...........414 9.12 Inverse Problems ......................................414 9.12.1 The Nature of the Image ........................414 9.12.2 Inverse Problems: Nonuniqueness .................416 9.12.3 Inverse Problems: Finiteness, Compactness .........418 Contents XVII 9.12.4 Uniqueness and Rigidity Results ..................419 9.12.4.1 Vignéras Surfaces .......................420 9.13 Special Cases ..........................................421 9.13.1 Riemann Surfaces ...............................421 9.13.2 Space Forms ...................................424 9.13.2.1 Scars ..................................426 9.14 The Spectrum of Exterior Differential Forms ..............426 10 Geodesic Dynamics ........................................431 10.1 Introduction ..........................................432 10.2 Some Well Understood Examples ........................436 10.2.1 Surfaces of Revolution ...........................436 10.2.1.1 Zoll Surfaces ...........................436 10.2.1.2 Weinstein Surfaces ......................440 10.2.2 Ellipsoids and Morse Theory .....................440 10.2.3 Flat and Other Tori: Influence of the Fundamental Group .........................................442 10.2.3.1 Flat Tori ...............................442 10.2.3.2 Manifolds Which are not Simply Connected 443 10.2.3.3 Tori, not Flat ...........................445 10.2.4 Space Forms ...................................446 10.2.4.1 Space Form Surfaces .....................446 10.2.4.2 Higher Dimensional Space Forms ..........448 10.3 Geodesies Joining Two Points ...........................449 10.3.1 Birkhoff s Proof for the Sphere ...................449 10.3.2 Morse Theory ..................................453 10.3.3 Discoveries of Morse and Serre ...................454 10.3.4 Computing with Entropy ........................456 10.3.5 Rational Homology and Gromov s Work ...........458 10.4 Periodic Geodesies .....................................461 10.4.1 The Difficulties .................................461 10.4.2 General Results ................................463 10.4.2.1 Gramoli and Meyer ......................463 10.4.2.2 Results for the Generic ( Bumpy ) Case ... 465 10.4.3 Surfaces .......................................466 10.4.3.1 The Lusternik-Schnirelmann Theorem .....466 10.4.3.2 The Bangert-Franks-Hingston Results .....468 10.5 The Geodesic Flow .....................................471 10.5.1 Review of Ergodic Theory of Dynamical Systems ... 471 10.5.1.1 Ergodicity and Mixing ...................471 10.5.1.2 Notions of Entropy ......................473 10.6 Negative Curvature ....................................478 10.6.1 Distribution of Geodesies ........................481 10.6.2 Distribution of Periodic Geodesies ................481 10.7 Nonpositive Curvature ..................................482 XVIII Contents 10.8 Entropies on Various Space Forms .......................483 10.8.1 Liouville Entropy ...............................485 10.9 From Osserman to Lohkamp ............................485 10.10 Manifolds All of Whose Geodesies are Closed ..............488 10.10.1 Definitions and Caution .........................488 10.10.2 Bott and Samelson Theorems ....................490 10.10.3 The Structure on a Given Sd and KPn ............492 10.11 Inverse Problems: Conjugacy of Geodesic Flows ............495 11 Best Metrics ...............................................499 11.1 Introduction and a Possible Approach ....................499 11.1.1 An Approach ...................................501 11.2 Purely Geometric Functionals ...........................503 11.2.1 Systolic Inequalities .............................503 11.2.2 Counting Periodic Geodesies .....................504 11.2.3 The Embolie Constant ..........................504 11.2.4 Diameter and Injectivity .........................505 11.3 Least curved ..........................................506 11.3.1 Definitions .....................................506 11.3.1.1 inf WRhd/a .............................506 11.3.1.2 Minimal Volume ........................507 11.3.1.3 Minimal Diameter .......................507 11.3.2 The Case of Surfaces ............................508 11.3.3 Generalities, Compactness, Finiteness and Equivalence ................................509 11.3.4 Manifolds with inf Vol (resp. inf ЦДЦ^/з, inf diam) = 0..................511 11.3.4.1 Circle Fibrations and Other Examples..... 511 11.3.4.2 Allof-Wallach s Type of Examples .........513 11.3.4.3 Nilmanifolds and the Converse: Almost Flat Manifolds ...................514 11.3.4.4 The Examples of Cheeger and Rong .......514 11.3.5 Some Manifolds with inf Vol > 0 and inf ЦиЦх/уа > 0.............................515 11.3.5.1 Using Integral Formulas ..................515 11.3.5.2 The Simplicial Volume of Gromov .........516 11.3.6 inf \|\|Ji\|\|jT,d/a in Four Dimensions ...................518 11.3.7 Summing up Questions on inf Vol, inf ЦДЦ^/а ......519 11.4 Einstein Manifolds .....................................520 11.4.1 Hilbert s Variational Principle and Great Hopes ___520 11.4.2 The Examples from the Geometric Hierarchy .......524 11.4.2.1 Symmetric Spaces .......................524 11.4.2.2 Homogeneous Spaces and Others ..........524 11.4.3 Examples from Analysis: Evolution by Ricci Flow ... 525 Contents XIX 11.4.4 Examples from Analysis: Kahler Manifolds .........526 11.4.5 The Sporadic Examples .........................528 11.4.6 Around Existence and Uniqueness ................529 11.4.6.1 Existence ..............................529 11.4.6.2 Uniqueness .............................530 11.4.6.3 Moduli ................................531 11.4.6.4 The Set of Constants, Ricci Flat Metrics ... 532 11.4.7 The Yamabe Problem ...........................533 11.5 The Bewildering Fractal Landscape of ПЅ (M) According to Nabutovsky ...............................534 12 From Curvature to Topology ...............................543 12.1 Some History, and Structure of the Chapter ...............543 12.1.1 Hopfs Inspiration ..............................543 12.1.2 Hierarchy of Curvatures .........................546 12.1.2.1 Control via Curvature ...................546 12.1.2.2 Other Curvatures .......................547 12.1.2.3 The Problem of Rough Classification ......548 12.1.2.4 References on the Topic, and the Significance of Noncompact Manifolds .....549 12.2 Pinching Problems .....................................549 12.2.1 Introduction ...................................549 12.2.2 Positive Pinching ...............................552 12.2.2.1 The Sphere Theorem ....................552 12.2.2.2 Sphere Theorems Invoking Bounds on Other Invariants .....................557 12.2.2.3 Homeomorphic Pinching .................558 12.2.2.4 The Sphere Theorem with Lower Bound on Diameter, and no Upper Bound on Curvature ...........................562 12.2.2.5 Topology at the Diameter Pinching Limit .. 565 12.2.2.6 Pointwise Pinching ......................567 12.2.2.7 Cutting Down the Hypotheses ............567 12.2.3 Pinching Near Zero .............................568 12.2.4 Negative Pinching ..............................569 12.2.5 Ricci Curvature Pinching ........................571 12.3 Curvature of Fixed Sign ................................576 12.3.1 The Positive Side: Sectional Curvature ............576 12.3.1.1 The Known Examples ...................576 12-3.1.2 Homology Type and the Fundamental Group ..............580 12-3.1.3 The Noncompact Case ...................583 12.3.1.4 Positivity of the Curvature Operator ......588 12.3.1.5 Possible Approaches, Looking to the Future 590 XX Contents 12.3.2 Ricci Curvature: Positive, Negative and Just Below . 593 12.3.3 The Positive Side: Scalar Curvature ...............599 12.3.3.1 The Hypersurfaces of Schoen & Yau .......600 12.3.3.2 Geometrical Descriptions .................601 12.3.3.3 Gromov s Quantization of .řf-theory and Topological Implications of Positive Scalar Curvature ..............602 12.3.3.4 Trichotomy .............................603 12.3.3.5 The Proof ..............................603 12.3.3.6 The Gromov-Lawson Torus Theorem ......604 12.3.4 The Negative Side: Sectional Curvature ............605 12.3.4.1 Introduction ............................605 12.3.4.2 Literature ..............................605 12.3.4.3 Quasi-isometries ........................606 12.3.4.4 Volume and Fundamental Group ..........609 12.3.4.5 Negative Versus Nonpositive Curvature .... 612 12.3.5 The Negative Side: Ricci Curvature ...............613 12.4 Finiteness and Collapsing ...............................614 12.4.1 Finiteness ......................................614 12.4.1.1 Cheeger s Finiteness Theorems ............614 12.4.1.2 More Finiteness Theorems ...............618 12.4.1.3 Ricci Curvature .........................622 12.4.2 Compactness and Convergence ...................624 12.4.2.1 Motivation .............................624 12.4.2.2 History ................................624 12.4.2.3 Contemporary Definitions and Results .....625 12.4.3 Collapsing and the Space of Riemannian Metrics. ... 630 12.4.3.1 Collapsing ..............................630 12.4.3.2 Closures on a Compact Manifold ..........634 13 Holonomy Groups and Kahler Manifolds ...................637 13.1 Definitions and Philosophy ..............................637 13.2 Examples .............................................639 13.3 General Structure Theorems ............................641 13.4 Classification ..........................................643 13.5 The Rare Cases ........................................646 13.5.1 G2 and Spin(7) .................................646 13.5.2 Quaternionic Kahler Manifolds ...................647 13.5.2.1 The Bérard Bergery/Salamon Twistor Space of Quaternionic Kahler Manifolds .... 649 13.5.2.2 The Konishi Twistor Space of a Quaternionic Kahler Manifold ........651 13.5.2.3 Other Twistor Spaces ....................651 13.5.3 Ricci Flat Kahler and Hyper-Kähler Manifolds .....652 13.5.3.1 Hyperkähler Manifolds ...................652 Contents XXI 13.6 Kahler Manifolds ......................................654 13.6.1 Symplectic Structures on Kahler Manifolds .........655 13.6.2 Imitating Complex Algebraic Geometry on Kahler Manifolds ............................655 14 Some Other Important Topics ..............................659 14.1 Noncompact Manifolds .................................660 14.1.1 Noncompact Manifolds of Nonnegative Ricci Curvature ...................660 14.1.2 Finite Volume ..................................661 14.1.3 Bounded Geometry .............................661 14.1.4 Harmonic Functions .............................662 14.1.5 Structure at Infinity .............................662 14.1.6 Chopping ......................................662 14.1.7 Positive Mass ..................................662 14.1.8 Cohomology and Homology Theories ..............663 14.2 Bundles over Riemannian Manifolds ......................663 14.2.1 Differential Forms and Related Bundles ............663 14.2.1.1 The Hodge Star .........................664 14.2.1.2 A Variational Problem for Differential Forms and the Laplace Operator ..........664 14.2.1.3 Calibration .............................666 14.2.1.4 Harmonic Analysis of Other Tensors .......667 14.2.2 Spinors ........................................668 14.2.2.1 Algebra of Spinors ......................668 14.2.2.2 Spinors on Riemannian Manifolds .........669 14.2.2.3 History of Spinors .......................669 14.2.2.4 Applying Spinors ........................670 14.2.2.5 Warning: Beware of Harmonic Spinors .....670 14.2.2.6 The Half Pontryagin Class ...............670 14.2.2.7 Reconstructing the Metric from the Dirac Operator .................671 14.2.2.8 Spin0 Structures ........................671 14.2.3 Various Other Bundles ..........................671 14.2.3.1 Secondary Characteristic Classes ..........672 14.2.3.2 Yang-Mills Theory ......................672 14.2.3.3 Twistor Theory .........................672 14.2.3.4 tf-theory ...............................673 14.2.3.5 The Atiyah-Singer Index Theorem ........674 14.2.3.6 Supersymmetry and Supergeometry .......674 14.3 Harmonic Maps Between Riemannian Manifolds ...........674 14.4 Low Dimensional Riemannian Geometry ..................676 14.5 Some Generalizations of Riemannian Geometry ............676 14.5.1 Boundaries .....................................676 14.5.2 Orbifolds ......................................677 XXII Contents 14.5.3 Conical Singularities ............................678 14.5.4 Spectra of Singular Spaces .......................678 14.5.5 Alexandrov Spaces ..............................678 14.5.6 CAT Spaces ....................................680 14.5.6.1 The CAT(fe) Condition ..................680 14.5.7 Camot-Carathéodory Spaces .....................681 14.5.7.1 Example: the Heisenberg Group ...........681 14.5.8 Finsler Geometry ...............................682 14.5.9 Riemannian Foliations ...........................683 14.5.10 Pseudo-Riemannian Manifolds ....................683 14.5.11 Infinite Dimensional Riemannian Geometry ........684 14.5.12 Noncommutative Geometry ......................685 14.6 Gromov s mm Spaces ..................................685 14.7 Submanifolds ..........................................690 14.7.1 Higher Dimensions ..............................690 14.7.2 Geometric Measure Theory and Pseudoholomorphic Curves ...................691 15 The Technical Chapter .....................................693 15.1 Vector Fields and Tensors ...............................693 15.2 Tensors Dual via the Metric: Index Aerobics ..............696 15.3 The Connection, Covariant Derivative and Curvature .......697 15.4 Parallel Transport .....................................701 15.4.1 Curvature from Parallel Transport ................703 15.5 Absolute (Ricci) Calculus and Commutation Formulas: Index Gymnastics ......................................704 15.6 Hodge and the Laplacian, Bochner s Technique ............706 15.6.1 Bochner s Technique for Higher Degree Differential Forms ..............708 15.7 Gaufr-Bonnet-Chern ...................................709 15.7.1 Chern s Proof of Gaufr-Bonnet for Surfaces ........710 15.7.2 The Proof of Allendoerfer and Weil ...............711 15.7.3 Chern s Proof in all Even Dimensions .............713 15.7.4 Chern Classes of Vector Bundles ..................714 15.7.5 Pontryagin Classes ..............................715 15.7.6 The Euler Class ................................715 15.7.7 The Absence of Other Characteristic Classes .......716 15.7.8 Applying Characteristic Classes ..................716 15.7.9 Characteristic Numbers ..........................716 15.8 Examples of Curvature Calculations ......................718 15.8.1 Homogeneous Spaces ............................718 15.8.2 Riemannian Submersions ........................719 Contents XXIII References .....................................................723 Acknowledgements ............................................789 List of Notation ...............................................791 List of Authors ................................................797 Subject Index .................................................811
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spelling	Berger, Marcel 1927-2016 Verfasser (DE-588)12047672X aut A panoramic view of Riemannian geometry Marcel Berger Berlin [u.a.] Springer 2003 XXIII, 824 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Riemannsche Geometrie Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009165941&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Berger, Marcel 1927-2016 A panoramic view of Riemannian geometry Riemannsche Geometrie Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd
subject_GND	(DE-588)4128462-8
title	A panoramic view of Riemannian geometry
title_auth	A panoramic view of Riemannian geometry
title_exact_search	A panoramic view of Riemannian geometry
title_full	A panoramic view of Riemannian geometry Marcel Berger
title_fullStr	A panoramic view of Riemannian geometry Marcel Berger
title_full_unstemmed	A panoramic view of Riemannian geometry Marcel Berger
title_short	A panoramic view of Riemannian geometry
title_sort	a panoramic view of riemannian geometry
topic	Riemannsche Geometrie Geometry, Riemannian Riemannsche Geometrie (DE-588)4128462-8 gnd
topic_facet	Riemannsche Geometrie Geometry, Riemannian
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009165941&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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