21-450: Topics in Geometry (Differential Geometry)
Differential Geometry stands at a crossroads in mathematics. It is the point
where topology, geometry and analysis combine. It is the language for many
fields of modern mathematics, and finds applications in physics, robotics and
In this course, we will introduce the basic concepts, and work to develop an
intuitive understanding of them. We begin with the basics. Chapters 1-3
review some of the background material and develop the definition of a
differentiable manifold. Chapters 3 and 4 introduce the idea of a tangent
vectors on a manifold, and go on to discuss the tangent bundle and other
types of tensor. Chapters 6 and 7 discuss calculus on manifolds,
the ideas of multivariable calculus and analysis in this new setting.
Chapter 8 discusses curvature, hilighting some classical results along the way.
While it will not be possible to cover the entire textbook, it is hoped that
a sufficient amount of each topic can be covered to give a good feeling for
the flavor of the subject.
The text for this course is An Introduction to Differentiable Manifolds
and Riemannian Geometry by William M. Boothby.
Out main text is a good "elementary" introduction that touches on all the
major points of the subject: differentiable manifolds, vector fields and tensor
fields, the tangent space and vector bundles, differentiation and integration
on manifolds, and various topics relating to curvature. However, there a few
other books to which I would like to call your attention.
- Calculus on Manifolds, by Michael Spivak. This is a very nice
little book. I expands on the material in Chapter 6 of Boothby. I would
recommend it as a companion text for that part of the course.
- Topology from the Differentiable Viewpoint, by John W. Milnor.
This, like Calculus on Manifolds is a small paperback. It would
provide satisfying reading to a student who, at the end of the course, was
interested in a quick overview and an introduction to some other
- A Comprehensive Introduction to Differential Geometry, Vol. 1,
by Michael Spivak. Encyclopedic. This book takes quite a bit of reading,
but if you make it through, you will really have a good understanding of the
material. I would highly recommend this for anyone entertaining
thoughts of graduate school in mathematics.
- Riemannian Geometry, by Manfredo Perdigao DoCarmo. A more
advanced book, I would recommend this to the student who at the end of the
course wants to continue their study of differential geometry.
Dr. David Handron
Office: WEH 6121