MA0106: Differential Geometry

Differential geometry is the study of objects of various dimensions, using the tools of differential calculus and linear algebra. In all parts of this course, we will be concerned with the concept of curvature, a notion that describes deviation from straightness or flatness. The structure of the course will emphasize the ways in which this central concept appears in all dimensions.

Lab 1: Local Theory of Plane Curves

↑ Introduction

1.1: Plane Curves and Their Representation

1.2: Velocity Vectors and Speed

1.3: Acceleration and Normal Vectors

1.4: Smooth Curves and Curves with Singular Points

1.5: Pedal Curves

Lab 2: Theory of Plane Curves, Continued

↑ Introduction

2.1: Decomposition of Acceleration

2.2: The Tangential Image

2.3: Parallel Curves

2.4: Evolutes and Osculating Circles

2.5: Distance Functions to Plane Curves

2.6: Reconstruction from Curvature

2.7: Global Theory of Plane Curves

Lab 3: Local Theory of Space Curves

↑ Introduction

3.1: Space Curves and Their Representation

3.2: Velocity Vectors and Speed

3.3: Unit Tangent and Principal Normal Vectors

3.4: The Binormal Vector and the Frenet Frame

3.5: The Frenet-Serret Equations

3.6: Curvature and Osculating Circles

3.7: Parallel Curves

Lab 4: Theory of Space Curves, Continued

↑ Introduction

4.1: Spherical Images of Space Curves

4.2: Distance Functions to Space Curves

4.3: Reconstruction from Curvature and Torsion

4.4: Pedal Curves

4.5: Involute Curves of Space Curves

4.6: Osculating Spheres of Space Curves

4.7: Global Theory of Space Curves

Lab 5: Local Theory of Surfaces

↑ Introduction

5.1: Surfaces and Their Representation

5.2: Velocity and Arc Length of Parameter Curves

5.3: Velocity and Arc Length of General Curves on Surfaces

5.4: Linear Independence in Terms of Metric Coefficients

5.5: Areas of Regions

5.6: Lengths And Areas Given Intrinsically

5.7: Angles between Parameter Curves

Lab 6: Extrinsic Theory of Surfaces

↑ Introduction

6.1: The Gauss Map

6.2: Introduction to Gaussian Curvature

6.3: Investigating Gaussian Curvature

6.4: Parallel Surfaces and Mean Curvature

6.5: Extra Material

Lab 7: The Normal Map and Gaussian Curvature

↑ Introduction

7.1: Evolute Surfaces

7.2: The Weingarten Map

7.3: Minimal Surfaces and Their Deformations

Lab 8: Normal and Geodesic Curvature

↑ Introduction

8.1: Normal Curvature of Curves on Surfaces

8.2: The Coefficients of the Second Fundamental Form

8.3: Principal Curvatures as Extrema of Normal Curvature

8.4: Elliptic, Parabolic, and Hyperbolic Points

8.5: Normal Sections

8.6: Meusnier's Theorem

8.7: Geodesic Curvature of Curves on Surfaces

8.8: Theorema Egregium

Lab 9: The Gauss-Bonnet Theorem

↑ Introduction

9.1: Total Curvature of a Surface

9.2: Normal Images of Tubes around Smooth Curves on Surfaces

9.3: Normal Images of Tubes around Piecewise Smooth Curves

9.4: The Euler Characteristic

9.5: Ptolemy's Theorem

9.6: The Upper Half-Plane

9.7: Geodesic Curvature in Orthogonal Coordinates

9.8: The Intrinsic Form of the Gauss-Bonnet Theorem