Differential Geometry

Duration

Four Weeks

Lecturer

Tanya Schmah (Macquarie University)

Consultation Hours:

Tuesday 12-13, 14-15 in Carslaw Room 609. (These are times when I promise to be there and available. Please feel free to make an appointment, or just drop by, at other times.

Assessment

80% final exam (Friday 9 Feb 9–11 in Carslaw Room 275), 20% assignments.

Assumed Knowledge

A good grounding in multivariable calculus, linear algebra and introductory real analysis, as well as some familiarity with ODEs and groups. Some experience of classical mechanics is desirable but not necessary.

Resources

There is an Assignment (PDF) part of which you are expected to hand in by Wednesday in the first week (not part of the assessment). See http://www.maths.mq.edu.au/~schmah/ice07/ for more resources.

Course Outline

This is an introduction to Riemannian and symplectic geometry on manifolds, with some applications to mechanics. We begin with the geometry of parametrised surfaces, including: Gaussian curvature; geodesics; isometries and area-preserving maps; and gradient and skew-gradient vector fields. The second part of the course is about manifolds, including: vector fields, tensors and forms; Lie derivatives; Frobenius’ theorem; orientation; and Lie groups. In the latter third of the course, we formally introduce Riemannian geometry, symplectic geometry and Hamiltonian mechanics. Time permitting, the course will conclude with an introduction to Lie group symmetries in mechanics.