Welcome to the wonderful world of Geometry!

This course is an introduction to differential and Riemannian geometry: a beautiful language in which much of modern mathematics and physics is spoken. We will address questions like what is the possible shape of the universe? What is a "straight line" in a general space? and what can it tell us about that space? We will cover

  • The geometry of Curves in Euclidean space,
  • Manifolds in Euclidean space,
  • The geometry of surfaces in Euclidean space,
  • Intrinsic geometry of surfaces,
  • Influences of curvature on topology.
Given time we will also discuss some (or all?) of the following topics:
  • Minimal surfaces and surfaces of constant mean curvature (Soap bubbles),
  • Analysis on surfaces,
  • Higher dimensional manifolds.

Announcements:

  • Please take part in the Course/Instructor Opinion Survey (CIOS). Written comments are most appreciated and will help shape future versions of this course (and others). Please comment on specific things you liked and did not like about the course. This current course has benefitted from past feedback, so please help improve the course further. Your feedback is most appreciated!
  • The final exam will be Friday, December 9th, from 2:50 to 5:40. The room is Skiles 171 (to be confirmed). The exam will be comprehesive covering all the materical in the class. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems, (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
    • The final exam will consist of 6 problems, plus one problem consisting of some true/false questions.
    • Most will be similar to the homework problems.
    • As a "practice test" work the questions on the previous two test, plus the practice problems for the tests and in addition work problems 1, 2, 3, and 4 on HW #5 and problems 1, 2, 3, and 4 on these practice problems.
    • I guarantee that at least 3 problem on the test will be either be among these problems or very similar to them. These "guaranteed problems" will be evenly distributed, that is at least one will be from the material covered by the first test, one from the second test and one from the material since the second test.
    • Some of the formulas from the later part of the course are complicated and not so easy to remember. Since I would like you to focus on understanding the meaning of the definitions and theorems and how to apply formulas, I do not want you to have to memorize these formulas. So this formulas sheet will be attached to your final exam. (So you do not have to have these formulas memorized, but you do need to know the various theorems and definitions from class and how to use the theorems and formulas. These are not necessary all the useful formulas from the class you might need, but just some of the basic ones.)
  • The second midterm exam will cover the material covered from section II.C through section IV in the notes and on the homework assignments 3 and 4. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems, (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
    • The test will be approximately 5 or 6 questions.
    • Most will be similar to the homework problems.
    • As a "practice test" work problems 1, 4, and 5 from homework 3 and problems 3, 4, 9, and 10 from homework 4.
    • I guarantee that at least 2 of the problems on the test will either be among these problems or very similar to them.
    • One of the questions on the test will consist of several True/False or short answer questions.
    • I will provide this formula sheet on the test. (So you do not have to have these formulas memorized, but you do need to know the various theorems and definitions from class and how to use the theorems and formulas. These are not necessary all the useful formulas from the class you might need, but just some of the basic ones.)
  • The second test in the class will be in class on November 7th.
  • The first midterm exam will cover the material covered through section II.B in the notes and on the homework assignments 1 and 2. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems, (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
    • The test will be approximately 5 or 6 questions.
    • Most will be similar to the homework problems.
    • As a "practice test" work problems 5, 7, and 11 from homework 1 and problems 1, 3, 9, and 11 from homework 2.
    • I guarantee that at least 2 of the problems on the test will either be among these problems or very similar to them.
    • One of the questions on the test will consist of several True/False or short answer questions.
    • I will provide this formula sheet on the test. (So you do not have to have these formulas memorized, but you do need to know the various theorems from class and how to use the theorems and formulas.)
  • The first test in the class will be in class on September 26th.

Course Information: