Ⅰ.Introduction to Manifolds

1.Preliminary Comments on Rn

2.Rn and Euclidean Space

3.Topological Manifolds

4.Further Examples of Manifolds Cutting and Pasting

5.Abstract Manifolds Some Examples

Ⅱ.Functions of Several Variables and Mappings

1.Differentiability for Functions of Several Variables

2.Differentiability of Mappings and Jacobians

3.The Space of Tangent Vectors at a Point of Rn

4.Another Definition of Ta （Rn）

5.Vector Fields on Open Subsets of Rn

6.The Inverse Function Theorem

7.The Rank of a Mapping

Ⅲ.Differentiable Manifolds and Submanifolds

1.The Definition of a Differentiable Manifold

2.Further Examples

3.Differentiable Functions and Mappings

4.Rank of a Mapping, Immersions

5.Submanifolds

6.Lie Groups

7.The Action of a Lie Group on a Manifold Transformation Groups

8.The Action of a Discrete Group on a Manifold

9.Covering Manifolds

Ⅳ Vector Fields on a Manifold

1.The Tangent Space at a Point of a Manifold

2.Vector Fields

3.One-Parameter and Local One-Parameter Groups Acting on a Manifold

4.The Existence Theorem for Ordinary Differential Equations

5.Some Examples of One-Parameter Groups Acting on a Manifold

6.One-Parameter Subgroups of Lie Groups

7.The Lie Algebra of Vector Fields on a Manifold

8.Frobenius’s Theorem

9.Homogeneous Spaces

Ⅴ Tensors and Tensor Fields on Manifolds

1.Tangent Covectors

Covectors on Manifolds

Covector Fields and Mappings

2.Bilinear Forms.The Riemannian Metric

3.Riemannian Manifolds as Metric Spaces

4.Partitions of Unity

Some Applications of the Partition of Unity

5.Tensor Fields

Tensors on a Vector Space

Tensor Fields

Mappings and Covariant Tensors

The Symmetrizing and Alternating Transformations

6.Multiplication of Tensors

Multiplication of Tensors on a Vector Space

Multiplication of Tensor Fields

Exterior Multiplication of Alternating Tensors

The Exterior Algebra on Manifolds

7.Orientation of Manifolds and the Volume Element

8.Exterior Differentiation

An Application to Frobenius’s Theorem

Ⅵ.Integration on Manifolds

1.Integration in Rn Domains of Integration

Basic Properties of the Riemann Integral

2.A Generalization to Manifolds

Integration on Riemannian Manifolds

3.Integration on Lie Groups

4.Manifolds with Boundary

5.Stokes’s Theorem for Manifolds

6.Homotopy of Mappings.The Fundamental Group

Homotopy of Paths and Loops.The Fundamental Group

7.Some Applications of Differential Forms.The de Rham Groups

The Homotopy Operator

8.Some Further Applications of de Rham Groups

The de Rham Groups of Lie Groups

9.Covering Spaces and Fundamental Group

Ⅶ.Differentiation on Riemannian Manifolds

1.Differentiation of Vector Fields along Curves in Rn

The Geometry of Space Curves

Curvature of Plane Curves

2.Differentiation of Vector Fields on Submanifolds of Rn

Formulas for Covariant Derivatives

？xp Y and Differentiation of Vector Fields

3.Differentiation on Riemannian Manifolds

Constant Vector Fields and Parallel Displacement

4.Addenda to the Theory of Differentiation on a Manifold

The Curvature Tensor

The Riemannian Connection and Exterior Differential Forms

5.Geodesic Curves on Riemannian Manifolds

6.The Tangent Bundle and Exponential Mapping.Normal Coordinates

7.Some Further Properties of Geodesics

8.Symmetric Riemannian Manifolds

9.Some Examples

Ⅷ.Curvature

1.The Geometry of Surfaces in E3

The Principal Curvatures at a Point of a Surface

2.The Gaussian and Mean Curvatures of a Surface

The Theorema Egregium of Gauss

3.Basic Properties of the Riemann Curvature Tensor

4.Curvature Forms and the Equations of Structure

5.Differentiation of Covariant Tensor Fields

6.Manifolds of Constant Curvature

Spaces of Positive Curvature

Spaces of Zero Curvature

Spaces of Constant Negative Curvature

REFERENCES

INDEX