A First Course in Differential Geometry Surfaces in Euclidean Space Instructor Solution Manual Solutions 1st Edition by Lyndon Woodward, John Bolton 1108582044 9781108582049

1 Curves in R[sup(n)]

1.1 Basic definitions

1.2 Arc length

1.3 The local theory of plane curves

1.4 Involutes and evolutes of plane curves [sup(†)]

1.5 The local theory of space curves

Exercises

2 Surfaces in R[sup(n)]

2.1 Definition of a surface

2.2 Graphs of functions

2.3 Surfaces of revolution

2.4 Surfaces defined by equations

2.5 Coordinate recognition

2.6 Appendix: Proof of three theorems [sup(†)]

Exercises

3 Tangent planes and the first fundamental form

3.1 The tangent plane

3.2 The first fundamental form

3.3 Arc length and angle

3.4 Isothermal parametrisations

3.5 Families of curves

3.6 Ruled surfaces

3.7 Area

3.8 Change of variables [sup(†)]

3.9 Coordinate independence [sup(†)]

Exercises

4 Smooth maps

4.1 Smooth maps between surfaces

4.2 The derivative of a smooth map

4.3 Local isometries

4.4 Conformal maps

4.5 Conformal maps and local parametrisations

4.6 Appendix 1: Some substantial examples [sup(†)]

4.7 Appendix 2: Conformal and isometry groups [sup(†)]

Exercises

5 Measuring how surfaces curve

5.1 The Weingarten map

5.2 Second fundamental form

5.3 Matrix of the Weingarten map

5.4 Gaussian and mean curvature

5.5 Principal curvatures and directions

5.6 Examples: surfaces of revolution

5.7 Normal curvature

5.8 Umbilics

5.9 Special families of curves

5.10 Elliptic, hyperbolic, parabolic and planar points

5.11 Approximating a surface by a quadric [sup(†)]

5.12 Gaussian curvature and the area of the image of the Gauss map [sup(†)]

Exercises

6 The Theorema Egregium

6.1 The Christoffel symbols

6.2 Proof of the theorem

6.3 The Codazzi–Mainardi equations

6.4 Surfaces of constant Gaussian curvature [sup(†)]

6.5 A generalisation of Gaussian curvature [sup(†)]

Exercises

7 Geodesic curvature and geodesics

7.1 Geodesic curvature

7.2 Geodesics

7.3 Differential equations for geodesics

7.4 Geodesics as curves of stationary length

7.5 Geodesic curvature is intrinsic

7.6 Geodesics on surfaces of revolution [sup(†)]

7.7 Geodesic coordinates [sup(†)]

7.8 Metric behaviour of geodesics [sup(†)]

7.9 Rolling without slipping or twisting [sup(†)]

Exercises

8 The Gauss–Bonnet Theorem

8.1 Preliminary examples

8.2 Regular regions, interior angles

8.3 Gauss–Bonnet Theorem for a triangle

8.4 Classification of surfaces

8.5 The Gauss–Bonnet Theorem

8.6 Consequences of Gauss–Bonnet

Exercises

9 Minimal and CMC surfaces

9.1 Normal variations

9.2 Examples and first properties

9.3 Bernstein’s Theorem

9.4 Minimal surfaces and harmonic functions

9.5 Associated families

9.6 Holomorphic isotropic functions

9.7 Finding minimal surfaces

9.8 The Weierstrass–Enneper representation

9.9 Finding I, II, N and K

9.10 Surfaces of constant mean curvature

9.11 CMC surfaces of revolution

9.12 CMC surfaces and complex analysis

9.13 Link with Liouville and sinh-Gordon equations

9.14 CMC spheres

Exercises

10 Hints or answers to some exercises

Index