- 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.
- Office: 665 PGH
- Office hours: M 10-10:50am, W 1-1:50pm,
or by appointment
- Email: climenha [at] math.uh.edu
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.
- Lectures: MWF, 12-12:50pm, C 102
- Textbook: Differential Geometry and
Topology: With a View to Dynamical Systems, by
Keith Burns and Marian Gidea.
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.
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.
- Vector calculus, preferably including the
inverse function theorem and implicit function
- Linear algebra from the abstract point of
view: axiomatic treatment of vector spaces and
- Basic point-set topology: continuity and
compactness in metric spaces, and more generally
in topological spaces. Quotient space
HW 1 (due Wed, Jan
HW 2 (due Mon, Feb
HW 3 (due Mon, Feb
HW 4 (due Fri, Mar
HW 5 (due Mon, Apr
HW 6 (due Fri, Apr
HW 7 (due Mon, Apr