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1.

図書

図書
Marcel Berger
出版情報: Providence, R.I. : American Mathematical Society, c2000  ix, 182 p. ; 26 cm
シリーズ名: University lecture series ; 17
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2.

図書

図書
M・ ベルジェ [ほか] 著 ; 山崎直美訳
出版情報: 東京 : シュプリンガー・フェアラーク東京, 1987.4  2冊 ; 21cm
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3.

図書

図書
Marcel Berger & Bernard Gostiaux
出版情報: Paris : Librairie Armand Colin, 1972  384 p. ; 24 cm
シリーズ名: Collection U. Série Mathématiques
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4.

図書

図書
Marcel Berger, Paul Gauduchon, Edmond Mazet
出版情報: Berlin : Springer-Verlag, 1971  vii, 251 p. ; 26 cm
シリーズ名: Lecture notes in mathematics ; 194
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5.

図書

図書
Marcel Berger ... [et al.] ; translated by Silvio Levy
出版情報: New York ; Tokyo : Springer-Verlag, c1984  viii, 266 p. ; 25 cm
シリーズ名: Problem books in mathematics / edited by K. Bencsáth and P.R. Halmos
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6.

図書

図書
edited by Gu Chaohao, M. Berger and R.L. Bryant
出版情報: Berlin ; Tokyo : Springer-Verlag, c1987  xii, 243 p. ; 25 cm
シリーズ名: Lecture notes in mathematics ; 1255
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7.

図書

図書
edited by Marcel Berger, Shingo Murakami, Takushiro Ochiai
出版情報: Tokyo : Kaigai Pub., 1983  ii, 194 p. ; 24 cm
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8.

図書

図書
Alain Lascoux, Marcel Berger
出版情報: Berlin : Springer-Verlag, 1970  vii, 83 p. ; 26 cm
シリーズ名: Lecture notes in mathematics ; 154
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9.

図書

図書
Marcel Berger, Bernard Gostiaux ; translated from the French by Silvio Levy
出版情報: New York ; Berlin ; Tokyo : Springer-Verlag, c1988  xii, 474 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 115
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目次情報: 続きを見る
Background
Differential Equations
Differentiable Manifolds
Partitions of Unity, Densities and Curves
Critical Points
Differential Forms
Integration of Differential Forms
Degree Theory
Curves: The Local Theory
Plane Curves: The Global Theory
A Guide to the Local Theory of Surfaces in R3
A Guide to the Global Theory of Surfaces
Bibliography
Index of Symbols and Notations
Index
Background
Differential Equations
Differentiable Manifolds
10.

図書

図書
Marcel Berger
出版情報: Berlin ; London : Springer, c2003  xxiii, 824 p ; 24 cm.
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目次情報: 続きを見る
Euclidean Geometry / 1:
Preliminaries / 1.1:
Distance Geometry / 1.2:
A Basic Formula / 1.2.1:
The Length of a Path / 1.2.2:
The First Variation Formula and Application to Billiards / 1.2.3:
Plane Curves / 1.3:
Length / 1.3.1:
Curvature / 1.3.2:
Global Theory of Closed Plane Curves / 1.4:
"Obvious" Truths About Curves Which are Hard to Prove / 1.4.1:
The Four Vertex Theorem / 1.4.2:
Convexity with Respect to Arc Length / 1.4.3:
Umlaufsatz with Corners / 1.4.4:
Heat Shrinking of Plane Curves / 1.4.5:
Arnol'd's Revolution in Plane Curve Theory / 1.4.6:
The Isoperimetric Inequality for Curves / 1.5:
The Geometry of Surfaces Before and After Gauß / 1.6:
Inner Geometry: a First Attempt / 1.6.1:
Looking for Shortest Curves: Geodesics / 1.6.2:
The Second Fundamental Form and Principal Curvatures / 1.6.3:
The Meaning of the Sign of K / 1.6.4:
Global Surface Geometry / 1.6.5:
Minimal Surfaces / 1.6.6:
The Hartman-Nirenberg Theorem for Inner Flat Surfaces / 1.6.7:
The Isoperimetric Inequality in <$>{\op E}^3<$> à la Gromov / 1.6.8:
Notes / 1.6.8.1:
Generic Surfaces / 1.7:
Heat and Wave Analysis in <$>{\op E}^2<$> / 1.8:
Planar Physics / 1.8.1:
Bibliographical Note / 1.8.1.1:
Why the Eigenvalue Problem? / 1.8.2:
Minimax / 1.8.3:
Shape of a Drum / 1.8.4:
A Few Direct Problems / 1.8.4.1:
The Faber-Krahn Inequality / 1.8.4.2:
Inverse Problems / 1.8.4.3:
Heat / 1.8.5:
Eigenfunctions / 1.8.5.1:
Relations Between the Two Spectra / 1.8.6:
Heat and Waves in <$>{\op E}^3<$>, <$>{\op E}^d<$> and on the Sphere / 1.9:
Euclidean Spaces / 1.9.1:
Spheres / 1.9.2:
Billiards in Higher Dimensions / 1.9.3:
The Wave Equation Versus the Heat Equation / 1.9.4:
Transition / 2:
Surfaces from Gauß to Today / 3:
Gauß / 3.1:
Theorema Egregium / 3.1.1:
The First Proof of Gauß's Theorema Egregium; the Concept of ds2 / 3.1.1.1:
Second Proof of the Theorema Egregium / 3.1.1.2:
The Gauß-Bonnet Formula and the Rodrigues-Gauß Map / 3.1.2:
Parallel Transport / 3.1.3:
Inner Geometry / 3.1.4:
Alexandrov's Theorems / 3.2:
Angle Corrections of Legendre and Gauß in Geodesy / 3.2.1:
Cut Loci / 3.3:
Global Surface Theory / 3.4:
Bending Surfaces / 3.4.1:
Bending Polyhedra / 3.4.1.1:
Bending and Wrinkling with Little Smoothness / 3.4.1.2:
Mean Curvature Rigidity of the Sphere / 3.4.2:
Negatively Curved Surfaces / 3.4.3:
The Willmore Conjecture / 3.4.4:
The Global Gauß-Bonnet Theorem for Surfaces / 3.4.5:
The Hopf Index Formula / 3.4.6:
Riemann's Blueprints / 4:
Smooth Manifolds / 4.1:
Introduction / 4.1.1:
The Need for Abstract Manifolds / 4.1.2:
Examples / 4.1.3:
Submanifolds / 4.1.3.1:
Products / 4.1.3.2:
Lie Groups / 4.1.3.3:
Homogeneous Spaces / 4.1.3.4:
Grassmannians over Various Algebras / 4.1.3.5:
Gluing / 4.1.3.6:
The Classification of Manifolds / 4.1.4:
Surfaces / 4.1.4.1:
Higher Dimensions / 4.1.4.2:
Embedding Manifolds in Euclidean Space / 4.1.4.3:
Calculus on Manifolds / 4.2:
Tangent Spaces and the Tangent Bundle / 4.2.1:
Differential Forms and Exterior Calculus / 4.2.2:
Examples of Riemann's Definition / 4.3:
Riemann's Definition / 4.3.1:
Hyperbolic Geometry / 4.3.2:
Products, Coverings and Quotients / 4.3.3:
Coverings / 4.3.3.1:
Symmetric Spaces / 4.3.4:
Classification / 4.3.5.1:
Rank / 4.3.5.2:
Riemannian Submersions / 4.3.6:
Gluing and Surgery / 4.3.7:
Gluing of Hyperbolic Surfaces / 4.3.7.1:
Higher Dimensional Gluing / 4.3.7.2:
Classical Mechanics / 4.3.8:
The Riemann Curvature Tensor / 4.4:
Discovery and Definition / 4.4.1:
The Sectional Curvature / 4.4.2:
Standard Examples / 4.4.3:
Constant Sectional Curvature / 4.4.3.1:
Projective Spaces <$>{\op KP}^n<$> / 4.4.3.2:
Hypersurfaces in Euclidean Space / 4.4.3.3:
A Naive Question: Does the Curvature Determine the Metric? / 4.5:
Any Dimension / 4.5.1:
Abstract Riemannian Manifolds / 4.6:
Isometrically Embedding Surfaces in <$>{\op E}^3<$> / 4.6.1:
Local Isometric Embedding of Surfaces in <$>{\op E}^3<$> / 4.6.2:
Isometric Embedding in Higher Dimensions / 4.6.3:
A One Page Panorama / 5:
Metric Geometry and Curvature / 6:
First Metric Properties / 6.1:
Local Properties / 6.1.1:
Hopf-Rinow and de Rham Theorems / 6.1.2:
Convexity and Small Balls / 6.1.2.1:
Totally Geodesic Submanifolds / 6.1.4:
Center of Mass / 6.1.5:
Examples of Geodesics / 6.1.6:
First Technical Tools / 6.1.7:
Second Technical Tools / 6.3:
Exponential Map / 6.3.1:
Space Forms / 6.3.1.1:
Nonpositive Curvature / 6.3.3:
Triangle Comparison Theorems / 6.4:
Bounded Sectional Curvature / 6.4.1:
Ricci Lower Bound / 6.4.2:
Philosophy Behind These Bounds / 6.4.3:
Injectivity, Convexity Radius and Cut Locus / 6.5:
Definition of Cut Points and Injectivity Radius / 6.5.1:
Klingenberg and Cheeger Theorems / 6.5.2:
Convexity Radius / 6.5.3:
Cut Locus / 6.5.4:
Blaschke Manifolds / 6.5.5:
Geometric Hierarchy / 6.6:
The Geometric Hierarchy / 6.6.1:
Rank 1 Symmetric Spaces / 6.6.1.1:
Measure Isotropy / 6.6.1.3:
Negatively Curved Space Forms in Three and Higher Dimensions / 6.6.1.4:
Mostow Rigidity / 6.6.2.2:
Classification of Arithmetic and Nonarithmetic Negatively Curved Space Forms / 6.6.2.3:
Volumes of Negatively Curved Space Forms / 6.6.2.4:
Higher Rank Symmetric Spaces / 6.6.3:
Superrigidity / 6.6.4.1:
Volumes and Inequalities on Volumes of Cycles / 6.6.5:
Curvature Inequalities / 7.1:
Bounds on Volume Elements and First Applications / 7.1.1:
The Canonical Measure / 7.1.1.1:
Volumes of Standard Spaces / 7.1.1.2:
The Isoperimetric Inequality for Spheres / 7.1.1.3:
Sectional Curvature Upper Bounds / 7.1.1.4:
Ricci Curvature Lower Bounds / 7.1.1.5:
Isoperimetric Profile / 7.1.2:
Definition and Examples / 7.1.2.1:
The Gromov-Bérard-Besson-Gallot Bound / 7.1.2.2:
Nonpositive Curvature on Noncompact Manifolds / 7.1.2.3:
Curvature Free Inequalities on Volumes of Cycles / 7.2:
Curves in Surfaces / 7.2.1:
Loewner, Pu and Blatter-Bavard Theorems / 7.2.1.1:
Higher Genus Surfaces / 7.2.1.2:
The Sphere / 7.2.1.3:
Homological Systoles / 7.2.1.4:
Inequalities for Curves / 7.2.2:
The Problem, and Standard Manifolds / 7.2.2.1:
Filling Volume and Filling Radius / 7.2.2.2:
Gromov's Theorem and Sketch of the Proof / 7.2.2.3:
Higher Dimensional Systoles: Systolic Freedom Almost Everywhere / 7.2.3:
Embolic Inequalities / 7.2.4:
The Unit Tangent Bundle / 7.2.4.1:
The Core of the Proof / 7.2.4.3:
Croke's Three Results / 7.2.4.4:
Infinite Injectivity Radius / 7.2.4.5:
Using Embolic Inequalities / 7.2.4.6:
Transition: The Next Two Chapters / 8:
Spectral Geometry and Geodesic Dynamics / 8.1:
Why are Riemannian Manifolds So Important? / 8.2:
Positive Versus Negative Curvature / 8.3:
Spectrum of the Laplacian / 9:
History / 9.1:
Motivation / 9.2:
Setting Up / 9.3:
X definition / 9.3.1:
The Hodge Star / 9.3.2:
Facts / 9.3.3:
Heat, Wave and Schrodinger Equations / 9.3.4:
The Principle / 9.4:
An Application / 9.4.2:
Some Extreme Examples / 9.5:
Square Tori, Alias Several Variable Fourier Series / 9.5.1:
Other Flat Tori / 9.5.2:
<$>{\op KP}^n<$> / 9.5.3:
Other Space Forms / 9.5.5:
Current Questions / 9.6:
Direct Questions About the Spectrum / 9.6.1:
Direct Problems About the Eigenfunctions / 9.6.2:
Inverse Problems on the Spectrum / 9.6.3:
First Tools: The Heat Kernel and Heat Equation / 9.7:
The Main Result / 9.7.1:
Great Hopes / 9.7.2:
The Heat Kernel and Ricci Curvature / 9.7.3:
The Wave Equation: The Gaps / 9.8:
The Wave Equation: Spectrum & Geodesic Flow / 9.9:
The First Eigenvalue / 9.10:
λ1 and Ricci Curvature / 9.10.1:
Cheeger's Constant / 9.10.2:
λ1 and Volume; Surfaces and Multiplicity / 9.10.3:
Kähler Manifolds / 9.10.4:
Results on Eigenfunctions / 9.11:
Distribution of the Eigenfunctions / 9.11.1:
Volume of the Nodal Hypersurfaces / 9.11.2:
Distribution of the Nodal Hypersurfaces / 9.11.3:
The Nature of the Image / 9.12:
Inverse Problems: Nonuniqueness / 9.12.2:
Inverse Problems: Finiteness, Compactness / 9.12.3:
Uniqueness and Rigidity Results / 9.12.4:
Vignéras Surfaces / 9.12.4.1:
Special Cases / 9.13:
Riemann Surfaces / 9.13.1:
Scars / 9.13.2:
The Spectrum of Exterior Differential Forms / 9.14:
Geodesic Dynamics / 10:
Some Well Understood Examples / 10.1:
Surfaces of Revolution / 10.2.1:
Zoll Surfaces / 10.2.1.1:
Weinstein Surfaces / 10.2.1.2:
Ellipsoids and Morse Theory / 10.2.2:
Flat and Other Tori: Influence of the Fundamental Group / 10.2.3:
Flat Tori / 10.2.3.1:
Manifolds Which are not Simply Connected / 10.2.3.2:
Tori, not Flat / 10.2.3.3:
Euclidean Geometry / 1:
Preliminaries / 1.1:
Distance Geometry / 1.2:
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