Contents
1 Curves and Surfaces in ${ℝ}^3$
1.1 Space Curves and Length
1.2 Osculating Circles
1.3 Frenet-Serret Equations
1.4 Surfaces
1.5 Curvatures
1.6 Minimal Surfaces
2 Manifolds
2.1 Manifolds
2.2 Functions
2.3 Vectors
2.4 Vector Bundles
2.5 Summation Convention
2.6 The Lie Bracket
3 Metrics and Connections
3.1 Riemannian Metrics
3.2 Quaternions and ${𝕊}^3$
3.3 Covariant Derivatives
3.4 Parallel Transport
3.5 Torsion
3.6 The Levi-Civita connection
4 Geodesics
4.1 Straight Lines
4.2 The Hyperbolic Plane
4.3 Length and Distance
4.4 Exponential Maps
5 Curvature
5.1 Symmetries and Identities
5.2 Hypersurfaces
5.3 Sectional Curvature
A Literature
B Linear Algebra
Cross product
Linear Atlases
Bilinear Forms
C Examples