Preface to the Revised Second Edition |
Preface to the Second Edition |
Preface to the First Edition |
Introduction to Manifolds / I.: |
Preliminary Comments on R[superscript n] / 1.: |
R[superscript n] and Euclidean Space / 2.: |
Topological Manifolds / 3.: |
Further Examples of Manifolds. Cutting and Pasting / 4.: |
Abstract Manifolds. Some Examples / 5.: |
Functions of Several Variables and Mappings / II.: |
Differentiability for Functions of Several Variables |
Differentiability of Mappings and Jacobians |
The Space of Tangent Vectors at a Point of R[superscript n] |
Another Definition of T[subscript a](R[superscript n]) |
Vector Fields on Open Subsets of R[superscript n] |
The Inverse Function Theorem / 6.: |
The Rank of a Mapping / 7.: |
Differentiable Manifolds and Submanifolds / III.: |
The Definition of a Differentiable Manifold |
Further Examples |
Differentiable Functions and Mappings |
Rank of a Mapping, Immersions |
Submanifolds |
Lie Groups |
The Action of a Lie Group on a Manifold. Transformation Groups |
The Action of a Discrete Group on a Manifold / 8.: |
Covering Manifolds / 9.: |
Vector Fields on a Manifold / IV.: |
The Tangent Space at a Point of a Manifold |
Vector Fields |
One-Parameter and Local One-Parameter Groups Acting on a Manifold |
The Existence Theorem for Ordinary Differential Equations |
Some Examples of One-Parameter Groups Acting on a Manifold |
One-Parameter Subgroups of Lie Groups |
The Lie Algebra of Vector Fields on a Manifold |
Frobenius's Theorem |
Homogeneous Spaces |
Tensors and Tensor Fields on Manifolds / V.: |
Tangent Covectors |
Covectors on Manifolds |
Covector Fields and Mappings |
Bilinear Forms. The Riemannian Metric |
Riemannian Manifolds as Metric Spaces |
Partitions of Unity |
Some Applications of the Partition of Unity |
Tensor Fields |
Tensors on a Vector Space |
Mappings and Covariant Tensors |
The Symmetrizing and Alternating Transformations |
Multiplication of Tensors |
Multiplication of Tensors on a Vector Space |
Multiplication of Tensor Fields |
Exterior Multiplication of Alternating Tensors |
The Exterior Algebra on Manifolds |
Orientation of Manifolds and the Volume Element |
Exterior Differentiation |
An Application to Frobenius's Theorem |
Integration on Manifolds / VI.: |
Integration in R[superscript n] Domains of Integration |
Basic Properties of the Riemann Integral |
A Generalization to Manifolds |
Integration on Riemannian Manifolds |
Integration on Lie Groups |
Manifolds with Boundary |
Stokes's Theorem for Manifolds |
Homotopy of Mappings. The Fundamental Group |
Homotopy of Paths and Loops. The Fundamental Group |
Some Applications of Differential Forms. The de Rham Groups |
The Homotopy Operator |
Some Further Applications of de Rham Groups |
The de Rham Groups of Lie Groups |
Covering Spaces and Fundamental Group |
Differentiation on Riemannian Manifolds / VII.: |
Differentiation of Vector Fields along Curves in R[superscript n] |
The Geometry of Space Curves |
Curvature of Plane Curves |
Differentiation of Vector Fields on Submanifolds of R[superscript n] |
Formulas for Covariant Derivatives |
[down triangle, open subscript x subscript p] Y and Differentiation of Vector Fields |
Constant Vector Fields and Parallel Displacement |
Addenda to the Theory of Differentiation on a Manifold |
The Curvature Tensor |
The Riemannian Connection and Exterior Differential Forms |
Geodesic Curves on Riemannian Manifolds |
The Tangent Bundle and Exponential Mapping. Normal Coordinates |
Some Further Properties of Geodesics |
Symmetric Riemannian Manifolds |
Some Examples |
Curvature / VIII.: |
The Geometry of Surfaces in E[superscript 3] |
The Principal Curvatures at a Point of a Surface |
The Gaussian and Mean Curvatures of a Surface |
The Theorema Egregium of Gauss |
Basic Properties of the Riemann Curvature Tensor |
Curvature Forms and the Equations of Structure |
Differentiation of Covariant Tensor Fields |
Manifolds of Constant Curvature |
Spaces of Positive Curvature |
Spaces of Zero Curvature |
Spaces of Constant Negative Curvature |
References |
Index |
Preface to the Revised Second Edition |
Preface to the Second Edition |
Preface to the First Edition |
Introduction to Manifolds / I.: |
Preliminary Comments on R[superscript n] / 1.: |
R[superscript n] and Euclidean Space / 2.: |