Differential Geometry
A proof-based course in the differential geometry of curves and surfaces that pairs every theorem with computation: intrinsic and extrinsic geometry and the angle excess of spherical triangles, regular parametrized curves, arc length as the integral of speed and reparametrization by arc length, curvature, torsion, the osculating circle and the Frenet–Serret equations with the fundamental theorem of curves, global theorems on closed plane curves including the rotation index, Hopf's theorem on turning tangents, the four-vertex theorem and the isoperimetric inequality, regular surfaces given by parametrizations, graphs, level sets and surfaces of revolution, with tangent planes, normals and orientability, the first fundamental form for lengths, angles and areas, isometries and the Mercator and Lambert map projections, the Gauss map, the self-adjoint shape operator, the second fundamental form and Meusnier's theorem, principal, Gaussian and mean curvature, umbilic and compact surfaces, and minimal surfaces such as the catenoid and helicoid as critical points of area, Christoffel symbols and Gauss's Theorema Egregium through Brioschi's formula, with the Codazzi–Mainardi equations and Bonnet's theorem, covariant derivatives, parallel transport, holonomy, geodesics and Clairaut's relation, the local and global Gauss–Bonnet theorems linking total curvature to the Euler characteristic, and a capstone on smooth manifolds, Riemannian metrics and curvature tensors that analyses the Schwarzschild geometry of general relativity through Flamm's paraboloid, with runnable Python code throughout.
MATH-430 Year 4 3 credits 12 lessons · 5 h 21 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Distinguish intrinsic from extrinsic geometric quantities, and relate the angle excess of geodesic triangles on a sphere to their area.
- Work with regular parametrized curves, prove that arc length is the integral of speed and is independent of parametrization, and reparametrize curves by arc length.
- Compute curvature, torsion and the Frenet–Serret frame of space curves in any parametrization, and prove the fundamental theorem of curves.
- Prove and apply global theorems on closed plane curves: the rotation index and Hopf's theorem, the four-vertex theorem and the isoperimetric inequality.
- Verify that sets are regular surfaces using parametrizations, graphs, level sets and surfaces of revolution, and compute tangent planes, normals and orientations.
- Use the first fundamental form to compute lengths, angles and areas on surfaces, and recognize isometries, conformal maps and equal-area maps.
- Compute the Gauss map, the shape operator and the second fundamental form, prove that the shape operator is self-adjoint, and apply Meusnier's theorem.
- Compute principal, Gaussian and mean curvatures, classify points of a surface, prove the umbilic and compact-surface theorems, and derive the minimal surface condition from the first variation of area.
- Derive the Christoffel symbols from the metric, prove the Theorema Egregium through Brioschi's formula, and use it to decide when surfaces can be isometric.
- Define covariant derivatives, parallel transport and geodesics intrinsically, solve the geodesic equations, and apply Clairaut's relation and holonomy computations.
- Prove the local Gauss–Bonnet theorem, derive the global theorem relating total curvature to the Euler characteristic, and apply both to triangles, caps and closed surfaces.
- Describe smooth manifolds and Riemannian metrics, compute Christoffel symbols and curvature from a metric, and analyse the Schwarzschild spatial geometry and the predictions of general relativity.
Course content
12 lessons · 5 h 21 min total · final exam
- 1 Why differential geometry: measuring shape from the inside 26 min · quiz
- 2 Parametrized curves, arc length and reparametrization 28 min · quiz
- 3 Curvature, torsion and the Frenet–Serret frame 26 min · quiz
- 4 Global properties of plane curves: rotation index, four-vertex theorem, isoperimetric inequality 24 min · quiz
- 5 Regular surfaces and tangent planes 28 min · quiz
- 6 The first fundamental form: lengths, angles and area 26 min · quiz
- 7 The Gauss map and the second fundamental form 27 min · quiz
- 8 Gaussian and mean curvature, principal curvatures and minimal surfaces 25 min · quiz
- 9 Gauss's Theorema Egregium and isometries 28 min · quiz
- 10 Geodesics and the covariant derivative 28 min · quiz
- 11 The Gauss–Bonnet theorem 25 min · quiz
- 12 Capstone: smooth manifolds, Riemannian metrics and the road to general relativity 30 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A proof-based course in the differential geometry of curves and surfaces that pairs every theorem with computation: intrinsic and extrinsic geometry and the angle excess of spherical triangles, regular parametrized curves, arc length as the integral of speed and reparametrization by arc length, curvature, torsion, the osculating circle and the Frenet–Serret equations with the fundamental theorem of curves, global theorems on closed plane curves including the rotation index, Hopf's theorem on turning tangents, the four-vertex theorem and the isoperimetric inequality, regular surfaces given by parametrizations, graphs, level sets and surfaces of revolution, with tangent planes, normals and orientability, the first fundamental form for lengths, angles and areas, isometries and the Mercator and Lambert map projections, the Gauss map, the self-adjoint shape operator, the second fundamental form and Meusnier's theorem, principal, Gaussian and mean curvature, umbilic and compact surfaces, and minimal surfaces such as the catenoid and helicoid as critical points of area, Christoffel symbols and Gauss's Theorema Egregium through Brioschi's formula, with the Codazzi–Mainardi equations and Bonnet's theorem, covariant derivatives, parallel transport, holonomy, geodesics and Clairaut's relation, the local and global Gauss–Bonnet theorems linking total curvature to the Euler characteristic, and a capstone on smooth manifolds, Riemannian metrics and curvature tensors that analyses the Schwarzschild geometry of general relativity through Flamm's paraboloid, with runnable Python code throughout.
Part of the Science stream, typically taken in year 4.