An Introduction to Differentiable Manifolds and Riemannian Geometry. 
Second edition. 
Series: Pure and Applied Mathematics 120
W.M. Boothby: 
Academic Press, Inc., Orlando, FL, 1986

Contents

I. Introduction to Manifolds

1. Preliminary Comments on Rn
2. Rn and Euclidean Space 4
3. Topological Manifolds 6
4. Further Examples of Manifolds. Cutting and 
   Pasting Abstract Manifolds. 
   Some Examples 14 

II. Functions of Several Variables and Mappings

1. Differentiability for Functions of Several Variables 20
2. Differentiability of Mappings and Jacobians 25 
3. The Space of Tangent Vectors at a Point of Rn 29 
4. Another Definition of Ta(Rn) 32
5. Vector Fields on Open Subsets of Rn
6. The Inverse Function Theorem 41
7. The Rank of a Mapping 46

III. Differentiable Manifolds and Submanifolds

1. The Definition of a Differentiable Manifold
2. Further Examples 59
3. Differentiable Functions and Mappings
4. Rank of a Mapping, Immersions 68
5. Submanifolds 74
6. Lie Groups 80
7. The Action of a Lie Group on a Manifold. Transformation Groups 
8. The Action of a Discrete Group on a Manifold 93
9. Covering Manifolds 98

IV. Vector Fields on a Manifold

1. The Tangent Space at a Point of a Manifold
2. Vector Field 113
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 
4. The Existence Theorem for Ordinary Differential Equations 127 
5. Some Examples of One-Parameter Groups Acting on a Manifold
6. One-Parameter Subgroups of Lie Groups 142
7. The Lie Algebra of Vector Fields on a Manifold
8. Frobenius's Theorem	153
9. Homogeneous Spaces 160

V. Tensors and Tensor Fields on Manifolds

1. Tangent Covectors 171
   Covectors on Manifolds 172
   Covector Fields and Mappings 174
2. Bilinear Forms. The Riemannian Metric
3. Riemannian Manifolds as Metric Spaces
4. Partitions of Unity 186
   Some Applications of the Partition of Unity
5. Tensor Fields	192
   Tensors on a Vector Space
   Tensor Fields	194
   Mappings and Covariant Tensors 195
   The Symmetrizing and Alternating Transformations
6. Multiplication of Tensors 199
   Multiplication of Tensors on a Vector Space 
   Multiplication of Tensor Fields 201
   Exterior Multiplication of Alternating Tensors
   The Exterior Algebra on Manifolds 206
7. Orientation of Manifolds and the Volume Element
8. Exterior Differentiation	212
   An Application to Frobenius's Theorem 177 

VI. Integration on Manifolds

1. Integration in Rn Domains of Integration 223
   Basic Properties of the Riemann Integral 224
2. A Generalization to Manifolds	229
   Integration on Riemannian Manifolds
3. Integration on Lie Groups 237
4. Manifolds with Boundary 243
5. Stokes's Theorem for Manifolds 251
6. Homotopy of Mappings. The Fundamental Group 258 
   Homotopy of Paths and Loops. The Fundamental Group
7. Some Applications of Differential Forms. 
   The de Rham Groups
   The Homotopy Operator 268
8. Some Further Applications of de Rham Groups
   The de Rham Groups of Lie Groups 276
9. Covering Spaces and Fundamental Group 280

VII. Differentiation on Riemannian Manifolds

1. Differentiation of Vector Fields along Curves in Rn
   The Geometry of Space Curves 292
   Curvature of Plane Curves 296
2. Differentiation of Vector Fields on Submanifolds of Rn
   Formulas for Covariant Derivatives 303
   Differentiation of Vector Fields 305
3. Differentiation on Riemannian Manifolds 308
   Constant Vector Fields and Parallel Displacement
4. Addenda to the Theory of Differentiation on a Manifold
   The Curvature Tensor 316
   The Riemannian Connection and Exterior Differential Forms
5. Geodesic Curves on Riemannian Manifolds 321
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates 
7. Some Further Properties of Geodesics 332
8. Symmetric Riemannian Manifolds 340
9. Some Examples 346

VIII. Curvature

1. The Geometry of Surfaces in E^3
   The Principal Curvatures at a Point of a Surface 359
2. The Gaussian and Mean Curvatures of a Surface 363
   The Theorema Egregium of Gauss 366
3. Basic Properties of the Riemann Curvature Tensor
4. Curvature Forms and the Equations of Structure
5. Differentiation of Covariant Tensor Fields 384
6. Manifolds of Constant Curvature 391
   Spaces of Positive Curvature 394
   Spaces of Zero Curvature 396
   Spaces of Constant Negative Curvature

REFERENCES
INDEX