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Panoramic View of Riemannian Geometry

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ISBN-10: 3540653171

ISBN-13: 9783540653172

Edition: 2003

Authors: Marcel Berger

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Description:

The author introduces the main topics of Riemannian geometry. Since a Riemannian manifold is a subtle object, appealing to highly non-natural concepts, the first three chapters introduce the various concepts and tools of Riemannian geometry in a natural and motivating way.
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Book details

Copyright year: 2003
Publisher: Springer
Publication date: 6/29/2007
Binding: Hardcover
Pages: 824
Size: 6.50" wide x 9.75" long x 2.00" tall
Weight: 3.256
Language: English

Euclidean Geometry
Preliminaries
Distance Geometry
A Basic Formula
The Length of a Path
The First Variation Formula and Application to Billiards
Plane Curves
Length
Curvature
Global Theory of Closed Plane Curves
"Obvious" Truths About Curves Which are Hard to Prove
The Four Vertex Theorem
Convexity with Respect to Arc Length
Umlaufsatz with Corners
Heat Shrinking of Plane Curves
Arnol'd's Revolution in Plane Curve Theory
The Isoperimetric Inequality for Curves
The Geometry of Surfaces Before and After Gau�
Inner Geometry: a First Attempt
Looking for Shortest Curves: Geodesics
The Second Fundamental Form and Principal Curvatures
The Meaning of the Sign of K
Global Surface Geometry
Minimal Surfaces
The Hartman-Nirenberg Theorem for Inner Flat Surfaces
The Isoperimetric Inequality in <$>{\op E}^3<$> � la Gromov
Notes
Generic Surfaces
Heat and Wave Analysis in <$>{\op E}^2<$>
Planar Physics
Bibliographical Note
Why the Eigenvalue Problem?
Minimax
Shape of a Drum
A Few Direct Problems
The Faber-Krahn Inequality
Inverse Problems
Heat
Eigenfunctions
Relations Between the Two Spectra
Heat and Waves in <$>{\op E}^3<$>, <$>{\op E}^d<$> and on the Sphere
Euclidean Spaces
Spheres
Billiards in Higher Dimensions
The Wave Equation Versus the Heat Equation
Transition
Surfaces from Gau� to Today
Gau�
Theorema Egregium
The First Proof of Gau�'s Theorema Egregium; the Concept of ds<sup>2</sup>
Second Proof of the Theorema Egregium
The Gau�-Bonnet Formula and the Rodrigues-Gau� Map
Parallel Transport
Inner Geometry
Alexandrov's Theorems
Angle Corrections of Legendre and Gau� in Geodesy
Cut Loci
Global Surface Theory
Bending Surfaces
Bending Polyhedra
Bending and Wrinkling with Little Smoothness
Mean Curvature Rigidity of the Sphere
Negatively Curved Surfaces
The Willmore Conjecture
The Global Gau�-Bonnet Theorem for Surfaces
The Hopf Index Formula
Riemann's Blueprints
Smooth Manifolds
Introduction
The Need for Abstract Manifolds
Examples
Submanifolds
Products
Lie Groups
Homogeneous Spaces
Grassmannians over Various Algebras
Gluing
The Classification of Manifolds
Surfaces
Higher Dimensions
Embedding Manifolds in Euclidean Space
Calculus on Manifolds
Tangent Spaces and the Tangent Bundle
Differential Forms and Exterior Calculus
Examples of Riemann's Definition
Riemann's Definition
Hyperbolic Geometry
Products, Coverings and Quotients
Products
Coverings
Homogeneous Spaces
Symmetric Spaces
Classification
Rank
Riemannian Submersions
Gluing and Surgery
Gluing of Hyperbolic Surfaces
Higher Dimensional Gluing
Classical Mechanics
The Riemann Curvature Tensor
Discovery and Definition
The Sectional Curvature
Standard Examples
Constant Sectional Curvature
Projective Spaces <$>{\op KP}^n<$>
Products
Homogeneous Spaces
Hypersurfaces in Euclidean Space
A Naive Question: Does the Curvature Determine the Metric?
Surfaces
Any Dimension
Abstract Riemannian Manifolds
Isometrically Embedding Surfaces in <$>{\op E}^3<$>
Local Isometric Embedding of Surfaces in <$>{\op E}^3<$>
Isometric Embedding in Higher Dimensions
A One Page Panorama
Metric Geometry and Curvature
First Metric Properties
Local Properties
Hopf-Rinow and de Rham Theorems
Products
Convexity and Small Balls
Totally Geodesic Submanifolds
Center of Mass
Examples of Geodesics
Transition
First Technical Tools
Second Technical Tools
Exponential Map
Rank
Space Forms
Nonpositive Curvature
Triangle Comparison Theorems
Bounded Sectional Curvature
Ricci Lower Bound
Philosophy Behind These Bounds
Injectivity, Convexity Radius and Cut Locus
Definition of Cut Points and Injectivity Radius
Klingenberg and Cheeger Theorems
Convexity Radius
Cut Locus
Blaschke Manifolds
Geometric Hierarchy
The Geometric Hierarchy
Space Forms
Rank 1 Symmetric Spaces
Measure Isotropy
Symmetric Spaces
Homogeneous Spaces
Constant Sectional Curvature
Negatively Curved Space Forms in Three and Higher Dimensions
Mostow Rigidity
Classification of Arithmetic and Nonarithmetic Negatively Curved Space Forms
Volumes of Negatively Curved Space Forms
Rank 1 Symmetric Spaces
Higher Rank Symmetric Spaces
Superrigidity
Homogeneous Spaces
Volumes and Inequalities on Volumes of Cycles
Curvature Inequalities
Bounds on Volume Elements and First Applications
The Canonical Measure
Volumes of Standard Spaces
The Isoperimetric Inequality for Spheres
Sectional Curvature Upper Bounds
Ricci Curvature Lower Bounds
Isoperimetric Profile
Definition and Examples
The Gromov-B�rard-Besson-Gallot Bound
Nonpositive Curvature on Noncompact Manifolds
Curvature Free Inequalities on Volumes of Cycles
Curves in Surfaces
Loewner, Pu and Blatter-Bavard Theorems
Higher Genus Surfaces
The Sphere
Homological Systoles
Inequalities for Curves
The Problem, and Standard Manifolds
Filling Volume and Filling Radius
Gromov's Theorem and Sketch of the Proof
Higher Dimensional Systoles: Systolic Freedom Almost Everywhere
Embolic Inequalities
Introduction
The Unit Tangent Bundle
The Core of the Proof
Croke's Three Results
Infinite Injectivity Radius
Using Embolic Inequalities
Transition: The Next Two Chapters
Spectral Geometry and Geodesic Dynamics
Why are Riemannian Manifolds So Important?
Positive Versus Negative Curvature
Spectrum of the Laplacian
History
Motivation
Setting Up
X definition
The Hodge Star
Facts
Heat, Wave and Schrodinger Equations
Minimax
The Principle
An Application
Some Extreme Examples
Square Tori, Alias Several Variable Fourier Series
Other Flat Tori
Spheres
<$>{\op KP}^n<$>
Other Space Forms
Current Questions
Direct Questions About the Spectrum
Direct Problems About the Eigenfunctions
Inverse Problems on the Spectrum
First Tools: The Heat Kernel and Heat Equation
The Main Result
Great Hopes
The Heat Kernel and Ricci Curvature
The Wave Equation: The Gaps
The Wave Equation: Spectrum & Geodesic Flow
The First Eigenvalue
�<sub>1</sub> and Ricci Curvature
Cheeger's Constant
�<sub>1</sub> and Volume; Surfaces and Multiplicity
K�hler Manifolds
Results on Eigenfunctions
Distribution of the Eigenfunctions
Volume of the Nodal Hypersurfaces
Distribution of the Nodal Hypersurfaces
Inverse Problems
The Nature of the Image
Inverse Problems: Nonuniqueness
Inverse Problems: Finiteness, Compactness
Uniqueness and Rigidity Results
Vign�ras Surfaces
Special Cases
Riemann Surfaces
Space Forms
Scars
The Spectrum of Exterior Differential Forms
Geodesic Dynamics
Introduction
Some Well Understood Examples
Surfaces of Revolution
Zoll Surfaces
Weinstein Surfaces
Ellipsoids and Morse Theory
Flat and Other Tori: Influence of the Fundamental Group
Flat Tori
Manifolds Which are not Simply Connected
Tori, not Flat
Space Forms
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