9780132125895

Differential Geometry of Curves and Surfaces

Manfredo P Do Carmo

This volume covers local as well as global differential geometry of curves and surfaces.

2-2

Regular Surfaces; Inverse Images of Regular Values

Exercises

p.65

2-3

Change of Parameters; Differentiable Functions on Surfaces

Exercises

p.80

2-4

The Tangent Plane; The Differential of a Map

Exercises

p.88

2-5

The First Fundamental Form; Area

Exercises

p.99

2-6

Orientation of Surfaces

Exercises

p.109

4-2

Isometries; Conformal Maps

Exercises

p.227

4-3

The Gauss Theorem and the Equations of Compatibility

Exercises

p.237

4-4

Parallel Transport; Geodesics

Exercises

p.260

4-5

The Gauss-Bonnet Theorem and Its Applications

Exercises

p.282

4-6

The Exponential Map; Geodesic Polar Coordinates

Exercises

p.294

4-7

Further Properties of Geodesics; Convex Neighborhoods

Exercises

p.305

5-2

The Rigidity of the Sphere

Exercises

p.323

5-3

Complete Surfaces Theorem of Hopf-Rinow

Exercises

p.335

5-4

First and Second Variations of Arc Length; Bonnet's Theorem

Exercises

p.354

5-5

Jacobi Fields and Conjugate Points

Exercises

p.368

5-6

Covering Spaces; The Theorems of Hadamard

Exercises

p.388

5-7

Global Theorems for Curves; The Fary-Milnor Theorem

Exercises

p.404

5-9

Jacobi's Theorems

Exercises

p.424

5-10

Abstract Surfaces; Further Generalizations

Exercises

p.443

5-11

Hilbert's Theorem

Exercises

p.454