Differential Geometry

Duration
 Four Weeks

Lecturer
 Tanya Schmah (Macquarie University)

Consultation Hours:
 Tuesday 1213, 1415 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 areapreserving maps; and gradient and
skewgradient 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.
