Did differential geometry undergo a notation change?

Question

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by $d\omega-\omega\wedge \omega$, because they assumed that $Ds_{\alpha}=\sum_{i, \beta}\Gamma^{\beta}_{\alpha i}du_{i}\otimes s_{\beta}, \omega^{\beta}_{\alpha}=\sum_{i}\Gamma^{\beta}_{ai}du^{i}$, hence their connection matrix $\omega$ is the transpose of what we used nowadays. In today's notation, we have $\Omega=d\omega+\omega\wedge \omega$ instead.

This notation transition is not so difficult, until one actually encounters calculations using the two sets of notation, and the "translation process" is not easy. For example, the torsion free condition $\nabla_{X}-\nabla_{Y}-\nabla_{[X,Y]}=0$ is equivalent to $d\theta^{i}-\theta^{j}\wedge \theta^{i}_{j}=0$ using a local coframe field and connection matrix $\theta^{i}_{j}$. While this may be well-known to experts in the field, it certainly caused confusion to a new guy like me when I tried to read old papers (for example, Chern's paper on Chern-Gauss-Bonnet theorem). When I was learning differential geometry, "modern style" textbooks (De Carmo, Taubes, John Lee, Jurgen Jost, etc.) seem contented not to introduce the old notation. This problem can be solved by extensive googling around and reading old-style textbooks (like Chern's Lectures on differential geometry), but it takes some time at least.

I want to ask historically, what is the reason for this notation change? If I am not mistaken, this is a change of notation as well as a change of philosophy. In Chern's textbook there are a lot of explicit complicated computations to prove a trivial result (like connection exists on any manifold), and one gets the feeling that differential geometry is close to some kind tensor analysis. But I doubt if any modern reader (say, of John Lee's book) will feel the same way. What has happened since 60s-70s?

Reference:

Milnor: Morse Theory, Characteristic Classes (appendix)

Chern: Lectures on Differential Geometry, Chapter 4-5.

Answer 1

I don't know of a systematic change in notation, but Chern definitely used a different convention than what most other differential geometers did. The decision stems from homogeneous spaces and deciding whether you want to work with left or right cosets. Most people use left cosets, which has the consequence that the action of the isotropy group on the fiber on the principal bundle of orthonormal frames is a right action. Chern chose to do it the other way around (just as Herstein does in his algebra text).

It should be noted that there is no standard notation used by everybody in differential geometry. We all have our individual preferences. My experience is that no matter what you do, you run into situations where your choice of notation doesn't work so well.

This is probably a false memory, but I remember my advisor asking me if I had created my own notation yet.

Someone who has really given this a lot of thought and designed an idiosyncratic but quite versatile notation is Roger Penrose.

My advice is to set up your own preferred conventions and learn to convert whatever notation is being used in a paper or text into yours. It is also good to be comfortable with working with either vector fields directly (which for me is easier to interpret geometrically) or differential forms (where calculations are often a lot easier). And although it is usually better to work with respect to an orthonormal frame of tangent vectors and the corresponding dual frame, it is also useful to be to work with arbitrary frames. And, if you want to do PDE estimates, you usually need to convert everything into local co-ordinates.

ADDED: In terms of explicit formulas for a matrix group, there are two possible choices for the Maurer-Cartan form, either $A^{-1}\,dA$ or $dA\,A^{-1}$, depending on whether you want to work with left or right invariant vector fields. Again, most people use left invariant vector fields and set $\Omega = A^{-1}\,dA$, so the Maurer-Cartan equations are $$ d\Omega + \Omega\wedge\Omega = 0. $$ If instead you define $\Omega = dA\,A^{-1}$, you get $$ d\Omega - \Omega\wedge\Omega = 0. $$