Entry requirements

60 credits in mathematics or physics, of which 30 credits in mathematics. Linear Algebra II. Several Variable Calculus, Several Variable Calculus M or Geometry and Analysis III.

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. The fundamental theorem of curves in the plane and in space.

Regular surfaces: Local coordinates. Tangent space. The derivative as a linear map. Critical and regular values. Smooth maps between surfaces. Vector fields. Orientability. The first fundamental form. Isometries.

Geometry of surfaces in 3-space: The Gauss map. The Weingarten map. Second fundamental form. Normal curvature. Principal curvatures. Gaussian curvature. Mean curvature. Minimal surfaces.

Geodesics: Existence and uniqueness. Geodesic curvature. The exponential map. Normal neighborhoods. Normal and normal polar coordinates. Gauss’s Theorema Egregium. Minding’s theorem. Covariant derivative. Parallel transport.

The Gauss–Bonnet theorem.

Overview of higher‑dimensional Riemannian geometry and some modern results.

Instruction

Lectures and problem solving sessions

Assessment

Oral examination at the end of the course (8 credits). Written and oral presentations (2 credits).

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.