MATH740-0101: Fundamental Concepts of Differential Geometry-Spring 2015 jmr

MATH740-0101: Fundamental Concepts of Differential Geometry-Spring 2015 jmr

Instructor:

Jonathan Rosenberg. You may call me at or reach me by email at jmr@math.umd.edu. I usually attend the Geometry/Topology Seminar (Mondays 3-4), the Representation Theory and Algebra Seminars (Mondays and Wednesdays 2-3), the Geometry and Physics RIT (Thursday 3:30-4:30), and the Mathematics Colloquium (Wednesday 3:15-4:15). My office hours are Mondays and Wednesdays after class, or by appointment.  The TA for the course is Tianyu Ma, email tma123@umd.edu.

Meetings:

MWF at 9 in MTH 1308.

Prerequisites:

MATH 405 (rigorous linear algebra), MATH 411 (advanced calculus), and MATH 730 (basic topology).

Text:

Manfredo do Carmo, Riemannian Geometry, 1992, translated by F. Flaherty, Birkhäuser, ISBN 978-0-8176-3490-2.

Extra References:

Some of the material at the beginning of the course is not in Do Carmo.  Here are a few other references you might find useful on the differential topology topics (first 3 weeks of the semester):

  1. John Milnor, Topology from the Differentiable Viewpoint, Univ. of Virginia Press, 1965, republished by Princeton Univ. Press, 1997.
  2. R. Bott and L. Tu, Differential forms in algebraic topology, Graduate Texts in Math., no. 82, Springer, 1982.
  3. Frank Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Co., 1971, republished as Graduate Texts in Math., no. 94, Springer, 1983.
  4. I. Madsen and J. Tornehave, From calculus to cohomology: de Rham cohomology and characteristic classes, Cambridge Univ. Press, 1997.

For the material toward the end of the course on comparison theorems in Riemannian geometry, aside from Do Carmo you can consult:

  1. lecture notes of Eschenburg, also see link in the Files tab.  The figures are in a separate file.
  2. Jeff Cheeger and David Ebin, Comparison Theorems in Riemannian Geometry, North Holland, 1975, republished by AMS Chelsea, 2008.
  3. Lecture notes on Riemannian geometry by Richard Bishop of the Univ. of Illinois.

Quick Summary:

Riemannian geometry was created by Bernhard Riemann in his remarkable 1854 Habilitationsschrift (available at the Riemann archive as paper #13 in the original German and as paper #20 in translation). It provides the mathematical underpinnings for most analysis on manifolds as well as for general relativity theory.

This course will be a basic graduate course in differential geometry, in other words, in the use of methods of differential calculus to study manifolds. (If you don't already know what a manifold is, that's the very first topic!) We will start with some foundational material: the inverse and implicit function theorems, immersions and submersions, submanifolds, ideas of transversality.  Then we will cover differential forms, integration on manifolds, Stokes's Theorem, and the Poincaré lemma.  After that we will get to Riemannian geometry proper: Riemannian metrics, frame bundles, connections, geodesics, curvature, Jacobi fields, geometry of submanifolds. After covering all these preliminaries we will get to some of the main theorems in global Riemmannian geometry, which relate curvature properties to topology, such as the Cartan-Hadamard Theorem and Myers's Theorem. If time permits we will try to give an introduction to Lie groups and homogeneous spaces.

Course Requirements

There will be regular graded homework assignments, a mid-term exam, and a final exam. If you are taking the course for credit, then you are expected to turn in the homework on time and to get a reasonable fraction of it correct.

Course Evaluations

Please fill out the course evaluation questionnaire at CourseEvalUM during the period April 28, 2015 to May 13, 2015.  The course evaluations are particularly important this year since the course has been revamped recently and we would like to get some feedback on how well it is working in the new format.

Files Tab:

Please remember to look at the Files tab in this website.  In it you will find some of the original classic papers in the subject (always instructive to read), as well as things like homework and exam problem solutions.

Course Requirements and Grading Policy:

Homework will be assigned, collected, and graded, usually once a week. Homework counts for 40% of the grade. The mid-term counts 20% and the final exam counts 40%. Grades will have the following rough meaning (modified by pluses and minuses, as appropriate):

  • A. Good command of the material, at the level of a PhD pass on the geometry part of the topology/geometry PhD qualifying exam.
  • B. Partial command of the material, at the level of an MA pass on the geometry part of the topology/geometry PhD qualifying exam.
  • C. Rather weak understanding of the material.
  • D, F. I hope these won't arise. Basically mean "hardly ever showed up, expended no effort at all".

It is your responsibility to turn the homework in on time and to show up for the exams.  If you must miss an exam, please contact Dr. Rosenberg immediately about making it up.

Attendance Policy, Academic Dishonesty, Disabilities:

I won't take attendance, but missing a lot of classes might affect your grade if you are near the borderline between two grades.  On the homework, it is OK to talk to your fellow students about the problems (in this regard you can use the Chat feature on the website), but you should write up the answers yourself.  On the exams, no collaboration is allowed, and you are expected to write out and sign the honor pledge:

I pledge on my honor that I have not given or received any unauthorized assistance on this assignment/examination. 

If you think you need accommodation for some disability, please talk to the instructor and we'll make suitable arrangements.

Course Summary:

Date Details
CC Attribution Non-Commercial This course content is offered under a CC Attribution Non-Commercial license. Content in this course can be considered under this license unless otherwise noted.