Autumn 2001, MWF 1:30-2:20
Math 547 is an introduction to the primary concepts and
techniques of Riemannian geometry:
Riemannian metrics and connections, geodesics, curvature, and Jacobi
fields. We will use these tools to
prove the fundamental theorems relating curvature and topology: the theorems
of Gauss-Bonnet, Cartan-Hadamard,
Bonnet, and Cartan-Ambrose-Hicks.
Text:
Riemannian
Manifolds: An Introduction to Curvature,
by John M. Lee.
Prerequisites: Math 544/5/6, Topology and Geometry of
Manifolds.
I have been asked about background reading
and preparation for the course. None is
needed beyond knowing the material from 544/5/6. (The second chapter of Riemannian Manifolds contains a
quick review of this prerequisite material, if you would like a reminder.) However,
if you are interested there are a couple of undergraduate texts you could read
recreationally as an introduction to some ideas and examples: Riemannian Geometry, by Frank Morgan,
both the 1993 and 1998 editions, and chapter 12 in The Shape of Space: How
to Visualize Surfaces and Three-Dimensional Manifolds, by Jeffrey R.
Weeks. The 1998 edition of Morgan's
book will be on reserve. There's also a Scientific American article, ``Fiber
Bundles and Quantum Theory'' by Bernstein and Phillips, Sci. Am. 245
(July 1981), pp. 122-127, which introduces the reader to Riemannian geometry on
surfaces then applies the ideas to physics. {Scientific American is
available in the Natural Sciences Library.)
The rest of
the sequence: We probably
will spend the first few weeks of 548 finishing Riemannian Manifolds. The syllabus for the rest of 548 and
549 is yet to be determined, but will include introductions to most or all of
the following: complex and almost complex
structures on manifolds; principal and associated bundles and connections;
characteristic classes; symplectic and contact geometry.