Course Catalogue

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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

Recommended Entry Requirements

3H Metric Spaces and Basic Topology (Maths4077)

Differential Geometry (Maths 4010)

Assessment

90% Examination, 10% Coursework.

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

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 to use, the Frenet-Serret formulae for both plane and space curves;
(b) calculate curvature and lengths of curves, and areas of surfaces, and be able to prove their reparametrization invariance properties;
(c) state, and use, the properties of differentiable maps between surfaces;
(d) recognize special classes of surfaces such as ruled and developable surfaces, and surfaces of revolution together with their basic properties;
(e) calculate the Gauss map and curvature properties of surfaces (principal curvatures, mean and Gaussian curvatures) and to appreciate their geometrical meanings;
(f) explain the proof of Gauss' Theorema Egregium;
(g) calculate geodesic curvature and to construct geodesics on surfaces;
(h) calculate, both in terms of the Gauss map and via total curvature, the rotation index of a closed plane curve;
(i) use the Gauss-Bonnet Theorem to calculate topological properties of surfaces.

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