Unit · year 3
MU-306 · Differential Geometry
Threads space24 lectures4 theorems
Curvature of curves and surfaces, and the theorems that make it intrinsic.
PREREQUISITES
Lectures
| L01 | Curves in Space: Parametrisation — |
| L02 | Arc Length and Reparametrisation — |
| L03 | Curvature and Torsion |
| L04 | The Frenet–Serret Frame |
| L05 | The Frenet–Serret Formulas |
| L06 | The Fundamental Theorem of Curves |
| L07 | Plane Curves and the Rotation Index — |
| L08 | Regular Surfaces and Charts — |
| L09 | The Tangent Plane and Differentials — |
| L10 | The First Fundamental Form |
| L11 | Lengths, Angles, and Areas on Surfaces |
| L12 | The Gauss Map and the Second Fundamental Form — |
| L13 | Normal and Principal Curvatures — |
| L14 | Gaussian and Mean Curvature |
| L15 | Surfaces of Revolution and Ruled Surfaces — |
| L16 | Isometries and Intrinsic Geometry |
| L17 | The Theorema Egregium |
| L18 | Consequences: Why Flat Maps Must Distort |
| L19 | Geodesics — |
| L20 | Geodesic Curvature and the Exponential Map — |
| L21 | Parallel Transport and Holonomy — |
| L22 | The Gauss–Bonnet Theorem: Local Form |
| L23 | The Global Gauss–Bonnet Theorem |
| L24 | Synthesis: Curvature Between Local and Global |
Theorems in this unit
T-105
The Frenet–Serret formulas
Curvature and torsion determine a space curve.
T-106
The first fundamental form
Lengths and angles on a surface from its metric.
T-107
Gauss's Theorema Egregium
Gaussian curvature is intrinsic to the surface.
T-108
The Gauss–Bonnet theorem
Total curvature is a topological invariant.