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620-427 |
Lecturer: Craig Westerland, 168 Richard Berry, phone: 8344 9712, email: C.Westerland@ms.unimelb.edu.au
Time and Location: Tue 10:00-12:00 Room 2.05 Physics Podium
Subject Outline
Differential Geometry is the study of "metric" properties of
geometric objects; that is, those that are related to a notion of size,
distance, and curvature. Where in topology, there is for instance
only one
sphere (and indeed, a sphere is the same as a cube), in geometry
spheres of different sizes are regarded as different. In
differential geometry, the notion of distance is encoded in local or
infinitesimal data (the metric), and thus is also closely related to
calculus and differential equations.
In this course, students will become familiar with the basic notions of Riemannian metrics and curvature, geodesics and concrete examples such as hypersurfaces in Euclidean space, Lie groups and homogeneous spaces. Two fundamental tools of global differential geometry covered are Cartan Hadamard for manifolds of non positive curvature and O'Neill's formula for the curvature of homogeneous spaces.
Main Topics
- Manifolds and smooth functions
- Vector fields and derivatives
- Curves and submanifolds
- Hypersurfaces in Euclidean space
- Second fundamental form and curvature
- Examples and applications in dimension three.
- Riemannian metrics
- Bundles and connections
- Riemannian curvature
- Spaces of constant curvature
- Exponential map and normal coordinates
- Jacobi flelds and first and second variation
- Cartan Hadamard theorem
- Lie groups and Lie algebras
- Symmetric and homogenous spaces
- O'Neill's submersion formula.
References
- M. P. doCarmo, Riemannian Geometry, Birkhäuser, 1992 ISBN: 3764334908.
- N. J. Hicks, Notes
on Differential Geometry, Princeton, 1965
Assessment
Assessment will be based on two assignments to be handed in during semester (worth 50%) and a final 3-hour exam at the end of semester (worth 50%).
The plaigiarism declaration is available here