Vaughn Climenhaga

Associate Professor
Department of Mathematics
University of Houston


Math 7352

Riemannian Geometry
Spring 2018


Announcements:
  • The final exam will take place in the regular lecture room from 11am-2pm on Monday, May 7 (the university-scheduled final exam time).
  • The midterm test will take place in class on Monday, March 5.

Instructor: Vaughn Climenhaga
  • Office: 665 PGH
  • Office hours:  M 10-10:50am, W 1-1:50pm, or by appointment
  • Email: climenha [at] math.uh.edu

Course information:
  • Lectures:  MWF, 12-12:50pm, C 102
  • Textbook:  Differential Geometry and Topology: With a View to Dynamical Systems, by Keith Burns and Marian Gidea. 
  • Course syllabus
This course is an introduction to the theory of smooth manifolds, with an emphasis on their geometry.  The first third of the course will cover the basic definitions and examples of smooth manifolds, smooth maps, tangent spaces, and vector fields.  Later in the semester we will use Euclidean, spherical, and hyperbolic geometry to introduce the notion of a Riemannian metric; we will study parallel transport, geodesics, the exponential map, and curvature.  Other topics will include Lie theory and differential forms, including exterior differentiation and Stokes theorem.

As the subtitle suggests, the textbook we will use discusses some applications and examples in dynamical systems that are connected to Riemannian geometry.  While these connections may occasionally be mentioned in lectures, they will not be the focus of the course: this is first and foremost a course in Riemannian geometry, which is targeted towards the associated preliminary exam for our PhD program.

There are no formal prerequisites beyond graduate standing, but I will expect students to be familiar with the following.
  1. Vector calculus, preferably including the inverse function theorem and implicit function theorem.
  2. Linear algebra from the abstract point of view: axiomatic treatment of vector spaces and linear maps.
  3. Basic point-set topology: continuity and compactness in metric spaces, and more generally in topological spaces. Quotient space constructions.
A little bit of abstract algebra would also be helpful since we will occasionally mention groups and group actions, but this is less essential than the above elements.

Homework

HW 1 (due Wed, Jan 31)
HW 2 (due Mon, Feb 12)
HW 3 (due Mon, Feb 26)
HW 4 (due Fri, Mar 23)
HW 5 (due Mon, Apr 2)
HW 6 (due Fri, Apr 20)
HW 7 (due Mon, Apr 30)