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This course introduces students the idea of curvature, and ultimately, to one of the jewels in undergraduate mathematics: the famous Gauss-Bonnet Theorem that establishes a fundamental link between the subjects of geometry and topology.

34 x 1 hr lectures and 12 x 1 hr tutorials in a semester

Mandatory Entry Requirements

3H Metric Spaces and Basic Topology (Maths4077)

Recommended Entry Requirements

4H: Differential Geometry (MATHS4101)

Assessment

90% Examination, 10% Coursework.

Reassessment

Resit available for MSc students.

The main aim of this course is the study of the curvature properties of curves and surfaces. In particular, it will involve the study of smooth curves and regular surfaces in R^3 from a differentiable point of view, using techniques from calculus and geometry. A key notion is that of curvature, and we explore this from several points of view; in fact, an important aim of the course is to understand what curvature really is, not just its computation. The theorem of Gauss-Bonnet enables one to recover a simple topological invariant of a surface by computing a suitable integral defined in terms of the Gaussian curvature.

By the end of this course students will be able to:
(a) prove, and use, the Frenet-Serret formulae for both plane and space curves;
(b) calculate curvature and lengths of curves, and areas of surfaces, and prove their reparametrization invariance properties;
(c) state, and use, the properties of differentiable maps between surfaces;

(d) state, and use, the definition of a manifold;
(e) recognize special classes of surfaces such as ruled and developable surfaces, and surfaces of revolution together with their basic properties;
(f) calculate the Gauss map and curvature properties of surfaces (principal curvatures, mean and Gaussian curvatures) and to appreciate their geometrical meanings;
(g) prove Gauss' Theorema Egregium;
(h) calculate geodesic curvature and construct geodesics on surfaces;

(i) construct lines of curvature on surfaces;

(j) use the exponential map and perform calculations in geodesic polar coordinates;

(k) calculate and understand the properties of, the parallel transport of vector fields, and Jacobi fields and their relationships to geodesics;

(l) calculate, both in terms of the Gauss map and via total curvature, the rotation index of a closed plane curve and prove other basic results on the topology of closed curves (for example, the Four-Vertex Theorem and Isoperimetric inequality);
(m) prove, and use, the Gauss-Bonnet Theorem.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.