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Euclidean Geometry | |

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Preliminaries | |

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Distance Geometry | |

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A Basic Formula | |

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The Length of a Path | |

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The First Variation Formula and Application to Billiards | |

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Plane Curves | |

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Length | |

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Curvature | |

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Global Theory of Closed Plane Curves | |

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"Obvious" Truths About Curves Which are Hard to Prove | |

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The Four Vertex Theorem | |

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Convexity with Respect to Arc Length | |

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Umlaufsatz with Corners | |

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Heat Shrinking of Plane Curves | |

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Arnol'd's Revolution in Plane Curve Theory | |

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The Isoperimetric Inequality for Curves | |

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The Geometry of Surfaces Before and After Gauï¿½ | |

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Inner Geometry: a First Attempt | |

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Looking for Shortest Curves: Geodesics | |

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The Second Fundamental Form and Principal Curvatures | |

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The Meaning of the Sign of K | |

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Global Surface Geometry | |

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Minimal Surfaces | |

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The Hartman-Nirenberg Theorem for Inner Flat Surfaces | |

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The Isoperimetric Inequality in <$>{\op E}^3<$> ï¿½ la Gromov | |

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Notes | |

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Generic Surfaces | |

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Heat and Wave Analysis in <$>{\op E}^2<$> | |

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Planar Physics | |

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Bibliographical Note | |

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Why the Eigenvalue Problem? | |

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Minimax | |

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Shape of a Drum | |

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A Few Direct Problems | |

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The Faber-Krahn Inequality | |

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Inverse Problems | |

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Heat | |

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Eigenfunctions | |

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Relations Between the Two Spectra | |

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Heat and Waves in <$>{\op E}^3<$>, <$>{\op E}^d<$> and on the Sphere | |

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Euclidean Spaces | |

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Spheres | |

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Billiards in Higher Dimensions | |

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The Wave Equation Versus the Heat Equation | |

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Transition | |

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Surfaces from Gauï¿½ to Today | |

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Gauï¿½ | |

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Theorema Egregium | |

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The First Proof of Gauï¿½'s Theorema Egregium; the Concept of ds<sup>2</sup> | |

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Second Proof of the Theorema Egregium | |

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The Gauï¿½-Bonnet Formula and the Rodrigues-Gauï¿½ Map | |

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Parallel Transport | |

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Inner Geometry | |

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Alexandrov's Theorems | |

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Angle Corrections of Legendre and Gauï¿½ in Geodesy | |

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Cut Loci | |

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Global Surface Theory | |

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Bending Surfaces | |

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Bending Polyhedra | |

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Bending and Wrinkling with Little Smoothness | |

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Mean Curvature Rigidity of the Sphere | |

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Negatively Curved Surfaces | |

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The Willmore Conjecture | |

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The Global Gauï¿½-Bonnet Theorem for Surfaces | |

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The Hopf Index Formula | |

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Riemann's Blueprints | |

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Smooth Manifolds | |

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Introduction | |

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The Need for Abstract Manifolds | |

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Examples | |

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Submanifolds | |

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Products | |

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Lie Groups | |

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Homogeneous Spaces | |

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Grassmannians over Various Algebras | |

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Gluing | |

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The Classification of Manifolds | |

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Surfaces | |

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Higher Dimensions | |

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Embedding Manifolds in Euclidean Space | |

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Calculus on Manifolds | |

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Tangent Spaces and the Tangent Bundle | |

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Differential Forms and Exterior Calculus | |

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Examples of Riemann's Definition | |

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Riemann's Definition | |

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Hyperbolic Geometry | |

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Products, Coverings and Quotients | |

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Products | |

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Coverings | |

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Homogeneous Spaces | |

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Symmetric Spaces | |

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Classification | |

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Rank | |

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Riemannian Submersions | |

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Gluing and Surgery | |

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Gluing of Hyperbolic Surfaces | |

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Higher Dimensional Gluing | |

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Classical Mechanics | |

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The Riemann Curvature Tensor | |

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Discovery and Definition | |

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The Sectional Curvature | |

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Standard Examples | |

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Constant Sectional Curvature | |

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Projective Spaces <$>{\op KP}^n<$> | |

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Products | |

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Homogeneous Spaces | |

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Hypersurfaces in Euclidean Space | |

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A Naive Question: Does the Curvature Determine the Metric? | |

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Surfaces | |

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Any Dimension | |

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Abstract Riemannian Manifolds | |

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Isometrically Embedding Surfaces in <$>{\op E}^3<$> | |

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Local Isometric Embedding of Surfaces in <$>{\op E}^3<$> | |

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Isometric Embedding in Higher Dimensions | |

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A One Page Panorama | |

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Metric Geometry and Curvature | |

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First Metric Properties | |

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Local Properties | |

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Hopf-Rinow and de Rham Theorems | |

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Products | |

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Convexity and Small Balls | |

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Totally Geodesic Submanifolds | |

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Center of Mass | |

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Examples of Geodesics | |

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Transition | |

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First Technical Tools | |

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Second Technical Tools | |

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Exponential Map | |

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Rank | |

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Space Forms | |

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Nonpositive Curvature | |

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Triangle Comparison Theorems | |

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Bounded Sectional Curvature | |

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Ricci Lower Bound | |

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Philosophy Behind These Bounds | |

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Injectivity, Convexity Radius and Cut Locus | |

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Definition of Cut Points and Injectivity Radius | |

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Klingenberg and Cheeger Theorems | |

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Convexity Radius | |

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Cut Locus | |

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Blaschke Manifolds | |

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Geometric Hierarchy | |

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The Geometric Hierarchy | |

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Space Forms | |

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Rank 1 Symmetric Spaces | |

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Measure Isotropy | |

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Symmetric Spaces | |

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Homogeneous Spaces | |

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Constant Sectional Curvature | |

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Negatively Curved Space Forms in Three and Higher Dimensions | |

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Mostow Rigidity | |

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Classification of Arithmetic and Nonarithmetic Negatively Curved Space Forms | |

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Volumes of Negatively Curved Space Forms | |

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Rank 1 Symmetric Spaces | |

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Higher Rank Symmetric Spaces | |

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Superrigidity | |

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Homogeneous Spaces | |

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Volumes and Inequalities on Volumes of Cycles | |

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Curvature Inequalities | |

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Bounds on Volume Elements and First Applications | |

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The Canonical Measure | |

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Volumes of Standard Spaces | |

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The Isoperimetric Inequality for Spheres | |

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Sectional Curvature Upper Bounds | |

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Ricci Curvature Lower Bounds | |

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Isoperimetric Profile | |

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Definition and Examples | |

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The Gromov-Bï¿½rard-Besson-Gallot Bound | |

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Nonpositive Curvature on Noncompact Manifolds | |

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Curvature Free Inequalities on Volumes of Cycles | |

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Curves in Surfaces | |

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Loewner, Pu and Blatter-Bavard Theorems | |

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Higher Genus Surfaces | |

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The Sphere | |

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Homological Systoles | |

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Inequalities for Curves | |

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The Problem, and Standard Manifolds | |

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Filling Volume and Filling Radius | |

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Gromov's Theorem and Sketch of the Proof | |

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Higher Dimensional Systoles: Systolic Freedom Almost Everywhere | |

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Embolic Inequalities | |

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Introduction | |

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The Unit Tangent Bundle | |

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The Core of the Proof | |

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Croke's Three Results | |

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Infinite Injectivity Radius | |

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Using Embolic Inequalities | |

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Transition: The Next Two Chapters | |

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Spectral Geometry and Geodesic Dynamics | |

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Why are Riemannian Manifolds So Important? | |

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Positive Versus Negative Curvature | |

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Spectrum of the Laplacian | |

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History | |

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Motivation | |

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Setting Up | |

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X definition | |

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The Hodge Star | |

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Facts | |

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Heat, Wave and Schrodinger Equations | |

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Minimax | |

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The Principle | |

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An Application | |

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Some Extreme Examples | |

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Square Tori, Alias Several Variable Fourier Series | |

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Other Flat Tori | |

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Spheres | |

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<$>{\op KP}^n<$> | |

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Other Space Forms | |

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Current Questions | |

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Direct Questions About the Spectrum | |

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Direct Problems About the Eigenfunctions | |

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Inverse Problems on the Spectrum | |

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First Tools: The Heat Kernel and Heat Equation | |

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The Main Result | |

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Great Hopes | |

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The Heat Kernel and Ricci Curvature | |

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The Wave Equation: The Gaps | |

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The Wave Equation: Spectrum & Geodesic Flow | |

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The First Eigenvalue | |

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ï¿½<sub>1</sub> and Ricci Curvature | |

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Cheeger's Constant | |

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ï¿½<sub>1</sub> and Volume; Surfaces and Multiplicity | |

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Kï¿½hler Manifolds | |

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Results on Eigenfunctions | |

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Distribution of the Eigenfunctions | |

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Volume of the Nodal Hypersurfaces | |

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Distribution of the Nodal Hypersurfaces | |

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Inverse Problems | |

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The Nature of the Image | |

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Inverse Problems: Nonuniqueness | |

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Inverse Problems: Finiteness, Compactness | |

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Uniqueness and Rigidity Results | |

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Vignï¿½ras Surfaces | |

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Special Cases | |

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Riemann Surfaces | |

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Space Forms | |

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Scars | |

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The Spectrum of Exterior Differential Forms | |

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Geodesic Dynamics | |

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Introduction | |

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Some Well Understood Examples | |

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Surfaces of Revolution | |

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Zoll Surfaces | |

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Weinstein Surfaces | |

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Ellipsoids and Morse Theory | |

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Flat and Other Tori: Influence of the Fundamental Group | |

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Flat Tori | |

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Manifolds Which are not Simply Connected | |

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Tori, not Flat | |

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Space Forms | |