Appendix C
Examples

Throughout this script certain examples are used repeatedly. Here we link them together.

Helix

• Definition: Equation (1.3)
• Arc-length parameterisation: Example 1.8
• Curvature: Example 1.12
• Torsion: Example 1.16
• Relation of curvature and torsion to radius and slope: Example 1.18

Helicoid

• Definition: Equation (1.19)
• First fundamental form: Example 1.22
• Surface normal: Example 1.23
• Normal curvature: Example 1.27
• Second fundamental form: Example 1.31
• Principal curvatures: Example 1.36
• Metric: Example 3.3
• Conformality: Example 3.14

Euclidean Space

• Atlas: Example 2.3
• Continuous differentiability: Example 2.14
• Tangent bundle: Example 2.27
• Lie bracket: Example 2.37
• Metric: Example 3.2
• Covariant derivative: Example 3.21
• Torsion: Example 3.42
• Metric-compatibility: Example 3.57
• Riemannian curvature: Example 5.2

Polar Coordinates

• Definition: Example 2.4
• Equivalence of tangent vectors: Example 2.17
• Ill-definition of vector field differentiating a vector field: Example 2.33
• Lie bracket: Example 2.35
• Covariant derivative: Example 3.30

Glued Circle

• Definition: Example 2.5
• Embedding in ${ℝ}^2$: Example 2.16
• Directional derivative: Example 2.21
• Tangent bundle: Example 2.28
• Tangent bundle in ${ℝ}^4$: Exercise 2.31

Stereographic Projection

• Definition of ${𝕊}^1$: Example 2.11
• Definition of ${𝕊}^n$: Exercise 2.12
• Height function in charts: Example 2.15
• Coordinate vector fields on ${𝕊}^1$: Example 2.19
• Derivative of height function: Example 2.20
• Pushforward on ${𝕊}^1$: Example 2.22
• Pushforward on ${𝕊}^2$: Exercise 2.23
• Metric on ${𝕊}^1$: Example 3.6
• Change of charts for the metric: Exercise 3.7
• Metric on ${𝕊}^2$: Exercise 3.8
• Conformality: Example 3.13
• Covariant derivative: Example 3.26
• Christoffel coefficients of ${𝕊}^1$: Example 3.32
• Christoffel coefficients of ${𝕊}^2$: Exercise 3.35
• Torsion-free: Example 3.43
• Riemannian curvature tensor: Example 5.4
• Ricci curvature tensor: Example 5.10
• Einstein manifold: Example 5.11
• Gauss curvature: Example 5.16
• Sectional curvature: Example 5.24

3-Sphere

• Trivialisations: Example 3.16
• Left invariant vector field: Example 3.18
• Left covariant derivative: Example 3.29
• Parallel transport: Example 3.38
• Torsion of left connection: Example 3.45
• Torsion of right connection: Example 3.46
• Torsion-free connection: Example 3.47
• Metric-compatibility: Example 3.51
• Levi-Civita connection: Example 3.58

Tangent Connection

• Definition: Example 3.26
• Christoffel coefficients: Example 3.31
• Definition for Riemannian immersions: Definition 3.59
• Geodesics of sphere: Example 4.9

Hyperbolic Plane

• Definition, Metric, Connection, Isometries, and Geodesics: Section 4.2
• Length of geodesics: Example 4.16
• Exponential map and normal chart: Example 4.32
• Distance function: Example 4.36
• Metric coefficients in normal chart: Example 4.38