Differential Geometry, or Frustrated Beyond Frustration

So yesterday, I came here and vented about how I was trying to prove a trivial mundane fact from Kobayashi and Nomizu and how – despite struggling for hours – I felt close but still rather hopeless.

Enter today.

My game plan originally had been to spend today doing some of the studies I’d been neglecting recently (namely, Clifford analysis and Differential Algebra) but I really didn’t want to leave that identity without a proof. Then, I hashed a new plan: One hour on that identity, then split the day between Clifford analysis and Differential Algebra. Pretty simple, right?

Wrong.

After spending some time running morning errands, I sat down with the identity and went to work with some new strategies. Instead of trying to argue a notation-heavy thing from an abstract existence point of view, I reached into my manifold $M$, picked out a point $p$ with neighborhood $U$ and gave $U$ a local coordinate system $u_1,\ldots, u_n$. Then, I went to town, hitting shit with Vector Fields and composition and compositions of Vector Fields of compositions of vector-valued functions of….

…and now, here I am. It’s been two hours (and some change) since I started that identity and still, no proof whatsoever. So that got me to thinking: What if I really don’t understand properties of vector fields at all? What if I’m not a differential geometer or a differential topologist or a differential algebraist? What if I’m a differential nothing-er? What if my career is over before it starts?

So I decided to turn to Google.

I did some random search involving the terms “vector fields” and “composition” and – lo and behold – I found something that’s actually meaningful. And – and – I realized what my problem is…

I’m using a book published in 1963. It’s a classic. My book is a first edition of a text that’s so classic its reprints are selling on Amazon for almost \$200. It’s a gem. But 1963 was 50 years ago.

Fifty. Years. Ago.

My book is archaic and reading it is getting me into trouble.

I know this because my Googling turned up this wonderful resource, and upon reading lecture seven, my answer was immediately clear.

Basically, I was attempting to prove the Pullback-Pushforward identity (given here in K&N notation): Given a manifold $M$, a diffeomorphism $\varphi:M\to M$, and a vector field $X\in \mathbf{X}(M)$, along with points $p,q\in M$ for which $p=\varphi(q)$, the induced homeomorphisms $\varphi^*:\mathbf{F}(M)\to \mathbf{F}(M)$ for which $(\varphi^*f)(p)=(f(\varphi(p))$ and $\varphi_*:\mathbf{X}(M)\to\mathbf{X}(M)$ for which $(\varphi_*X)(p)=(\varphi_*)_q(X_q)$ interact in the following way:

$\varphi^*\left((\varphi_*X)f\right)=X(\varphi^*f)$ for all $X\in\mathbf{X}(M)$ and all $f\in\mathbf{F}(M)$.

Simple, right? Apparently the relationship I was missing is given in equations $(7.2)$ and $(7.3)$ in the seventh lecture linked above.

So now I have a dilemma.

I need to study Differential Geometry this summer. I’ve already invested almost a week – maybe a little more – in going through 10 pages of this book and trying to hash out the details. Now, I realize that with the archaic notation, 50 pages by summer’s end would be a good pace for someone as average as myself and how that’s just not good enough.

Clearly, newer documents are easier to read: The notation is cleaner and more intuitive and the exposition is aimed more at educating readers than simply expounding upon one’s own knowledge of the subject. Clearly I would have better luck progressing through documents like that.

Clearly, those are better for me.

But then it feels like a loss somehow, ya know? Like I started at the bottom of a mountain, scratched and clawed my way up through arduous paths and unfriendly conditions, and have come to a point where someone’s waiting in a helicopter to take me back up the hill to safety…

…and okay, then: My metaphor just answered my own question!

I guess tomorrow – or Saturday – or whenever – I’ll start studying Differential Geometry using some other textbook not written in the 60s (which means I’ll also probably skip out on Auslander’s book, too).

Until next time….

4 thoughts on “Differential Geometry, or Frustrated Beyond Frustration”

1. Muphrid

Being interested in Clifford algebra and analysis, you might find the work of David Hestenes and Garret Sobczyk interesting. They call their brand of Clifford algebra, very suggestively, *geometric algebra*, for they consistently emphasize geometric interpretations of elements of the algebra. There’s a nice chapter in their book about “vector manifolds” and the use of Clifford algebra (and an associated calculus!) to simplify and revamp the language of differential geometry.

Hestenes is a physicist however (as am I) and as such I fear he doesn’t make quite enough contact with the notation and concepts that a mathematician might be more familiar with. Geometric algebra is presented somewhat differently from the wider subject of Clifford algebra also, but you may find it interesting.

• Thanks for the resource suggestion, Muphrid! Up until now, I wasn’t familiar with those authors; I’ve since added their text to my Amazon cart for later investigation.

I was, however, somewhat familiar (though vaguely so) with the fields of geometric algebra and the calculus associated therewith. In particular, Alan Macdonald has a pair of books (see here and here), and I’m planning to buy them ASAP. I’d also actually just spoken with a professor (the same one doing the Clifford work) regarding resources in geometric algebra, and from what I gather, he’ll be sharing sources as well.