Math 7352
Riemannian Geometry
Spring 2018
Announcements:
 The final exam will take place in the regular
lecture room from 11am2pm on Monday, May 7
(the universityscheduled final exam time).
 The midterm test will take place in class
on Monday, March 5.
Instructor: Vaughn
Climenhaga
 Office: 665 PGH
 Office hours: M 1010:50am, W 11:50pm,
or by appointment
 Email: climenha [at] math.uh.edu
Course information:
 Lectures: MWF, 1212:50pm, 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.
 Vector calculus, preferably including the
inverse function theorem and implicit function
theorem.
 Linear algebra from the abstract point of
view: axiomatic treatment of vector spaces and
linear maps.
 Basic pointset 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)
