Differential Geometry 1

I was invited to sit in on a Graduate School course in differential geometry in the fall and I’m really excited. I’ve perused the field, studying a book about symmetry in mechanics that employed differential forms to simplify physics problems. There I was introduced to ideas of coordinate charts and atlases and tangent bundles, but I haven’t taken the time to dig directly in. I also haven’t studied math with someone else in years, so having a peer group will be fun. I think it will be online, but it’s at CUNY which is only a few blocks from my place, so I’ll have no problem masking up and walking over if they are open.

You may have a couple of questions. Like what is differential geometry and why am I studying it? Differential Geometry is calculus on hard mode. Instead of everything being on an x-y plane of real numbers, Diff Geo does everything on abstract topological spaces. I know, I just gave you an answer that induces more questions.

Let me break down what I know from a high level, starting with ordinary calculus. In ordinary calculus, you are introduced to 2 concepts: the derivative and the integral.

The derivative is rate at which one thing changes with another: how much distance is covered over some interval of time, how long your commute is as the number of people on the road changes or how many people die from an infectious disease as the number of people infected changes. If you can write the relationship between your dependent and independent quantities down as a formula, taking a derivative, or “differentiating” gives you a formula for the sensitivity to change at each point. This is equivalently stated as the “slope of a tangent line to a curve at a point.”

The integral, instead of giving you a rate, gives you an amount accumulated as a certain quantity varies. If you know how some quantity depends on another and you want to pinpoint the exact size then you integrate. Say you know that the national budget deficit grows at some number/hour and you want to figure out how much the deficit increased in a day, then you would take the integral of that rate over a 24 hour period. This is equivalently stated as the “area under the curve”

So calculus, conceptually, is all about moving between rates of change and absolute quantities. The derivative produces a rate of change, and integral produces an amount. Calculus extends by asking what happens in higher dimensions. Say the death rate is dependent on two quantities instead of one: number of infected people and amount of masks used. The basic rules stay the same but the added structure of extra dimensions makes for some fun techniques and interpretations. Multivariable calculus, as advanced as it may seem, comprises a fundamental set of skills needed for pretty much all science done at a professional, academic, or industrial level.

Differential geometry, as far as I can tell, adds the snag that your functions don’t live on flat worlds. They live on various types of “smooth manifolds”. Some examples of these smooth manifolds are spheres, or doughnuts, or projective spaces (projective space is a fun rabbit hole if you have free time). We can also do calculus on more exotic, higher dimensional manifolds that are hard to draw and kind of give you a headache to think about. Conceptually we still want to calculate rate of change or accumulated quantities from functions, but our computational techniques need to be generalized for novel situations. What does a function even mean in some strange twisted multidimensional space? What would its “rate of change” or “accumulated value” indicate? This is what I get to find out! You use this kind of thing in theoretical physics or mathematical biology or engineering. Anything where you are looking to calculate how a function might behave in the presence of curvature or multidimensionality.

But I’m learning it because its fun! My brushes with the subject have revealed tremendous beauty to me and I look forward to digging in.

I’ll chime in here and teach you guys differential geometry as I go about learning it. Chronicling my methods, my struggles and my aha’s.

Thanks to Nick Hillier for the Photo