Differential Geometry

Entry requirements

60 credits in mathematics (or 40 credits in mathematics and 40 credits in physics) including Several Variable Calculus M and Linear Algebra II. Several Variable Calculus M may be replaced by Geometry and Analysis III or Several Variable Calculus.

Learning outcomes

On completion of the course, the student should be able to:

• give an account of important differential geometric concepts and definitions,
• formulate and explain the meaning of important results and theorem,
• describe the main features of the proofs of central theorem and perform proof of simpler differential geometry,
• use the theory, methods and techniques of the course to solve mathematical problems.

Content

Regular curves: Arc length parametrization. Curvature and torsion. The Frenet frame and the Frenet equations. Regular surfaces: The derivative as a linear map. Critical and regular values. Local coordinates. Smooth maps between surfaces. Vector fields and covector fields. Geometry of surfaces in 3-space: First and second fundamental forms. The Gauss map. Normal curvature. Principal curvatures. Gaussian curvature. Codazzi-Mainardi equations and Bonnet's theorem. Theorema Egregium. Intrinsic geometry: Conformal maps and local isometries. Models for hyperbolic geometry. Covariant derivative, parallel transport. Geodesics. Geodesic curvature. Geodesic and normal coordinates. The exponential map. Minding's Theorem. Divergence and the Laplacian.

The Gauss-Bonnet Theorem. Overview of higher dimensional Riemannian geometry and some modern results.

Instruction

Lectures and problem solving sessions

Assessment

Assignments combined with oral examination.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.