447. Differential Geometry and
Topology
Wed 11 AM Maxwell
Wed 2PM Synge
Thur 9 AM Maxwell
Examination: Tue May 30 9:30-12:30
Regent House
The course is an introduction to differential geometry and topology at
the advanced undergraduate/beggining graduate level.
The course is recommended to mathematicians and to those theoretical
physicists interested in gravitational physics and gauge theories.
The course is concerned with mathematical results, and not with physical applications.
The following topics will be covered:
(1) Basic differential geometry (manifolds, differential structures,
differentiable maps,
vector and tensor fields, differential forms, Cartan calculus, Lie
group actions, orientability, integration on manifolds, de Rham
cohomology)
(2) Basic Riemannian geometry (metrics, Levi-Civita connection,
curvature tensor, basic Hodge theory)
(3) Fiber bundles (principal and vector bundles, connections,
characteristic classes, Chern-Weyl theory)
(4) Elements of algebraic topology (homology and cohomology theories,
applications to singular, Chech and de Rham theories).
Warning: The course assumes a
certain amount of mathematical maturity, in particular a thorough
understanding of analysis, multilinear algebra, group theory, general
topology and basic category theory.
Core textbook:
(1) Shigeyuki Morita, Geometry of
Differential Forms, American Mathematical Society (2001) ISBN:
0821810456
Recommended:
The treatment of some topics is inspired by the following books:
(1) Raoul Bott, Loring W. Tu, Differential Forms
in
Algebraic Topology, Springer (1995) ISBN: 0387906134
(2) Norman Steenrod, The Topology of Fibre Bundles, Princeton
University Press (1999) ISBN: 0691005486
(3) John Milnor, James D. Stasheff, Characteristic Classes, Princeton
University Press (1974) ISBN: 0691081220
Other classic references:
(4) Shoshichi Kobayashi, Katsumi Nomizu, Foundations
of Differential Geometry, vols 1,2;
Wiley-Interscience; New edition (1996) ISBN: 0471157333
(5) Michael Spivak, A comprehensive introduction to differential
geometry
vols 1-5, Publish or Perish, Inc; 3rd edition (1999) ISBN 0914098705
(6) Dale Husemoller, Fibre Bundles, Springer; 3rd edition (1993) ISBN:
0387940871
Special topics:
(7) John Milnor, Topology from the Differentiable Viewpoint, Princeton
University Press; Revised edition (1997)
ISBN: 0691048339
(8) Glen E. Bredon, Topology and Geometry, Springer (1997) ISBN:
0387979263
(9) J. P. May A Concise Course in Algebraic Topology, University Of
Chicago Press (1999)ISBN: 0226511839
Note: It is assumed that
students have a good understanding of certain results in multivariate
analysis.
The following reference might help in that regard:
Michael Spivak, Calculus on Manifolds: A Modern Approach to Classical
Theorems of Advanced Calculus,
HarperCollins Publishers (1965) ISBN: 0805390219