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1.

図書

図書
Luther Pfahler Eisenhart
出版情報: Princeton : Princeton Univ. Press, [n. d.].  vii, 306 p. ; 24 cm
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2.

図書

図書
Jürgen Jost
出版情報: Berlin : Springer, c2002  xiii, 532 p. ; 24 cm
シリーズ名: Universitext
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目次情報: 続きを見る
Foundational Material / 1:
Manifolds and Di erentiable Manifolds / 1.1:
Tangent Spaces / 1.2:
Submanifolds / 1.3:
Riemannian Metrics / 1.4:
Vector Bundles / 1.5:
Integral Curves of Vector Fields. Lie Algebras / 1.6:
Lie Groups / 1.7:
Spin Structures / 1.8:
Exercises for Chapter 1
De Rham Cohomology and Harmonic Dierential Forms / 2:
The Laplace Operator / 2.1:
Representing Cohomology Classes by Harmonic Forms / 2.2:
Generalizations / 2.3:
Exercises for Chapter 2
Parallel Transport, Connections, and Covariant Derivatives / 3:
Connections in Vector Bundles / 3.1:
Metric Connections. The Yang-Mills Functional / 3.2:
The Levi-Civita Connection / 3.3:
Connections for Spin Structures and the Dirac Operator / 3.4:
The Bochner Method / 3.5:
The Geometry of Submanifolds. Minimal Submanifolds / 3.6:
Exercises for Chapter 3
Geodesics and Jacobi Fields / 4:
1st and 2nd Variation of Arc Length and Energy / 4.1:
Jacobi Fields / 4.2:
Conjugate Points and Distance Minimizing Geodesics / 4.3:
Riemannian Manifolds of Constant Curvature / 4.4:
The Rauch Comparison Theorems and Other Jacobi Field Estimates / 4.5:
Geometric Applications of Jacobi Field Estimates / 4.6:
Approximate Fundamental Solutions and Representation Formulae / 4.7:
The Geometry of Manifolds of Nonpositive Sectional Curvature / 4.8:
Exercises for Chapter 4
A Short Survey on Curvature and Topology
Symmetric Spaces and Kahler Manifolds / 5:
Complex Projective Space. De nition of Kahler Manifolds / 5.1:
The Geometry of Symmetric Spaces / 5.2:
Some Results about the Structure of Symmetric Spaces / 5.3:
The Space Sl(n; R)=SO(n; R) / 5.4:
Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds / 5.5:
Exercises for Chapter 5
Morse Theory and Floer Homology / 6:
Preliminaries: Aims of Morse Theory / 6.1:
Compactness: The Palais-Smale Condition and the Existence of Saddle Points / 6.2:
Local Analysis: Nondegeneracy of Critical Points, Morse Lemma, Stable and Unstable Manifolds / 6.3:
Limits of Trajectories of the Gradient Flow / 6.4:
The Morse-Smale-Floer Condition: Transversality and Z2-Cohomology / 6.5:
Orientations and Z-homology / 6.6:
Homotopies / 6.7:
Graph ows / 6.8:
Orientations / 6.9:
The Morse Inequalities / 6.10:
The Palais-Smale Condition and the Existence of Closed Geodesics / 6.11:
Exercises for Chapter 6
Variational Problems from Quantum Field Theory / 7:
The Ginzburg-Landau Functional / 7.1:
The Seiberg-Witten Functional / 7.2:
Exercises for Chapter 7
Harmonic Maps / 8:
De nitions / 8.1:
Twodimensional Harmonic Mappings and Holomorphic Quadratic Di erentials / 8.2:
The Existence of Harmonic Maps in Two Dimensions / 8.3:
De nition and Lower Semicontinuity of the Energy Integral / 8.4:
Weakly Harmonic Maps. Regularity Questions / 8.5:
Higher Regularity / 8.6:
Formulae for Harmonic Maps. The Bochner Technique / 8.7:
Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Existence / 8.8:
Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Regularity / 8.9:
Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Uniqueness and Other properties / 8.10:
Exercises for Chapter 8
Linear Elliptic Partial Dierential Equation / Appendix A:
Sobolev Spaces / A.1:
Existence and Regularity Theory for Solutions of Linear Elliptic Equations / A.2:
Appendix
Fundamental Groups and Covering Spaces
Index
Foundational Material / 1:
Manifolds and Di erentiable Manifolds / 1.1:
Tangent Spaces / 1.2:
3.

図書

図書
translated from russian by Vladislav V. Goldberg, foreword by S.S. Chern
出版情報: New Jersey : World Scientific, c2001  xvii, 259 p. ; 23 cm
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4.

図書

図書
Olivier Druet, Emmanuel Hebey, Frédéric Robert
出版情報: Princeton, N.J. : Princeton University Press, c2004  viii, 218 p. ; 24 cm
シリーズ名: Mathematical notes ; 45
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5.

図書

図書
П.К. Рашевски
出版情報: Москва : "Наука" Глав. ред. физико-математической лит-ры, 1967  664 p. ; 22 cm
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6.

図書

図書
by Luther Pfahler Eisenhart
出版情報: Princeton : Princeton Univ. Press, c1926  vii, 262 p. ; 24 cm
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7.

図書

図書
Isaac Chavel
出版情報: Cambridge [England] : Cambridge University Press, 1995  xii, 386 p. ; 23 cm
シリーズ名: Cambridge tracts in mathematics ; 108
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目次情報: 続きを見る
Preface to the Second Edition
Preface
Riemannian Manifolds / I:
Connections / I.1:
Parallel Translation of Vector Fields / I.2:
Geodesics and the Exponential Map / I.3:
The Torsion and Curvature Tensors / I.4:
Riemannian Metrics / I.5:
The Metric Space Structure / I.6:
Geodesics and Completeness / I.7:
Calculations with Moving Frames / I.8:
Notes and Exercises / I.9:
Riemannian Curvature / II:
The Riemann Sectional Curvature / II.1:
Riemannian Submanifolds / II.2:
Spaces of Constant Sectional Curvature / II.3:
First and Second Variations of Arc Length / II.4:
Jacobi's Equation and Criteria / II.5:
Elementary Comparison Theorems / II.6:
Jacobi Fields and the Exponential Map / II.7:
Riemann Normal Coordinates / II.8:
Riemannian Volume / II.9:
Geodesic Spherical Coordinates / III.1:
The Conjugate and Cut Loci / III.2:
Riemannian Measure / III.3:
Volume Comparison Theorems / III.4:
The Area of Spheres / III.5:
Fermi Coordinates / III.6:
Integration of Differential Forms / III.7:
Appendix: Eigenvalue Comparison Theorems / III.8:
Riemannian Coverings / IV:
The Fundamental Group / IV.1:
Volume Growth of Riemannian Coverings / IV.3:
Discretization of Riemannian Manifolds / IV.4:
The Free Homotopy Classes / IV.5:
Surfaces / IV.6:
Systolic Inequalities / V.1:
Gauss-Bonnet Theory of Surfaces / V.2:
The Collar Theorem / V.3:
The Isoperimetric Problem: Introduction / V.4:
Surfaces with Curvature Bounded from Above / V.5:
The Isoperimetric Problem on the Paraboloid of Revolution / V.6:
Isoperimetric Inequalities (Constant Curvature) / V.7:
The Brunn-Minkowski Theorem / VI.1:
Solvability of a Neumann Problem in R[superscript n] / VI.2:
Fermi Coordinates in Constant Sectional Curvature Spaces / VI.3:
Spherical Symmetrization and Isoperimetric Inequalities / VI.4:
M. Gromov's Uniqueness Proof - Euclidean and Hyperbolic Space / VI.5:
The Isoperimetric Inequality on Spheres / VI.6:
The Kinematic Density / VI.7:
The Differential Geometry of Analytical Dynamics / VII.1:
The Berger-Kazdan Inequalities / VII.2:
On Manifolds with No Conjugate Points / VII.3:
Santalo's Formula / VII.4:
Isoperimetric Inequalities (Variable Curvature) / VII.5:
Croke's Isoperimetric Inequality / VIII.1:
Buser's Isoperimetric Inequality / VIII.2:
Isoperimetric Constants / VIII.3:
Discretizations and Isoperimetry / VIII.4:
Comparison and Finiteness Theorems / VIII.5:
Preliminaries / IX.1:
H. E. Rauch's Comparison Theorem / IX.2:
Comparison Theorems with Initial Submanifolds / IX.3:
Refinements of the Rauch Theorem / IX.4:
Triangle Comparison Theorems / IX.5:
Convexity / IX.6:
Center of Mass / IX.7:
Cheeger's Finiteness Theorem / IX.8:
Hints and Sketches for Exercises / IX.9:
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
Preface to the Second Edition
Preface
Riemannian Manifolds / I:
8.

図書

図書
Peter Petersen
出版情報: New York : Springer, c1998  xvi, 432 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 171
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目次情報: 続きを見る
Introduction
Riemannian Metrics
Curvature
Examples
Hypersurfaces
Geodesics and Distance
Sectional Curvature Comparison I
The Bochner Technique
Symmetric Spaces and Holony
Ricci Curvature Comparison
Convergence
Sectional Curvature Comparison II
Appendices
DeRham Cohomology / A:
Principal Bundles / B:
Spinors / C:
Bibliography
Introduction
Riemannian Metrics
Curvature
9.

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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10.

図書

図書
Thierry Aubin
出版情報: Berlin : Springer-Verlag, 1998  xvii, 395 p. ; 25 cm
シリーズ名: Springer monographs in mathematics
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