Math 670: Introduction to Differentiable Manifolds





WHEN: MWF 11 – 11:50 am

WHERE: WEBER 223 // in the oval with good weather

WHAT: A graduate level introduction to the theory of differentiable manifolds.

TEXTBOOK: Introduction to Smooth Manifolds (John M. Lee)




Since our wee years as budding mathematicians we have been fond and comfortable with euclidean spaces, i.e. spaces that can be coordinatized by n-tuples of real numbers and metrized using the Pythagorean theorem. We learned to study functions from euclidean spaces to euclidean spaces, and do all sort of fancy stuff such as differentiating and integrating. If we stop and ponder, however, we soon realize that to take a derivative, we don’t need the whole universe to be euclidean, but only a little open neighborhood od the point we want to take the derivative of. Integration is a global operation, but it is obtained by “adding up infinitely many infinitesimal things” (oh gods of math forgive my sloppiness) that are also constructed locally.

Therefore any space that can be locally (and appropriately) identified with euclidean space is suitable for integration and differentiation.

This is the basic insight that starts the theory of differentiable manifolds. Manifolds are precisely spaces that are “locally euclidean” but not necessarily globally so. Think this is a funky idea? Well if you think about it we live on a differentiable manifold, and we look at a collection of euclidean identifications of it everytime we flip through pages of the AAA Road Map of the US. So needless to say I think this is an extremely natural and foundational theory to just about all of mathematics (yes, even you purist of algebra and you applied mathematician should care about manifolds!)

The focus of this class will be on getting a basic overview of many features of the theory, introducing manifolds, bundles, vector fields and differential forms. We will prove some interesting non-trivial theorems, such as the Whitney embedding theorem. And we’ll at least introduce some interesting examples of manifolds, such as Lie Groups and manipulations thereof (such as Projective Spaces and Grassmannians). I am planning on covering good part of the first 10 chapters of the textbook. My guess is that this will already keep us entertained for most of the semester. 


HOMEWORK: Yes, there will be homework. Not much, but hopefully steady. Not necessarily graded, but probably collected. All sort of flexibility can be worked out, but in order to really make things sink in we should really make a (collective) effort to keep up with it.



Jan 28th

1) Exercises 1.1, 1.3 page 21.

2) Prove that a product of two smooth manifolds can be given the structure of a smooth manifold. Describe a possible smooth atlas for the torus.

Feb 4th

Let f(x,y) and g(x,y) be two polynomials in two variables. By setting

z = f(x,y)

w = g(x,y)

we get a smooth function from the plane (with coordinates x,y) to the plane (with coordinates z,w).

What are the conditions on f and g for this function to descend to a smooth function from the projective line to the projective line?

Can you generalize this construction to more than two variables and give  an example of a smooth function from P^3 to P^5?

Feb 11th

Exercises 3-1, page 64.

Feb 18th

Exercises 3-4, 3-5 page 64.

Feb 25th

Prove that the following conditions are equivalent:

1)     V is a smooth vector field on X

2)     Every component of every local representation of V is a smooth function.

3)     For every smooth function f: X->IR , the function Vf : X-> IR is smooth.

Mar 4th

Exercise 4.1, page 90

Mar 11th

Exercises 5-1, 5-5, 5-18 pages 126, 128

Apr  8th

Exercises 6-4,6-5 page 144

May 6th

Compute the de Rham cohomology vecor spaces of:

·        The n-dimensional sphere.

·        Real projective plane.

·        The figure eight.

·        The union at a point of two manifolds.

For the last two spaces, what we intend is “some manifold which is homotopy equivalent to…” but you can use the singular space in your constructions.