>>11208021Hey I didn't even do implicit summation in that example.
>>11208021I remember reading into Peterson Peterson at one point, but it didn't feel very much like physics, actually.
The General Relativity book by Wald
(
https://www.amazon.com/General-Relativity-Robert-M-Wald/dp/0226870332)
is a lot of Riemannian geometry next to the physics of it. I think by Peterson being physics you may mean dwelling around on various tensors for too long? And Laplacians in many explicit metrics?
>>11208025>if there's actually something more to differential geometry than studying manifoldsI'm not sure that that's what you really mean to say. Yes it's about manifolds, but there's lots of aspects to it.
I think for a physicist, or let's say without any ring theory at hand, it's initially tough to appreciate deRham cohomology. People like to present coordinate free Stokes, but without the whole algebra of the operators, looking back, this strikes me of a bit as a gimmick.
I don't know what topics you've seen or liked being tackled by an explicit differential geometric approach. There's this book by Waldmann (not sure if it has an English translation) which has 200-300 pages on the Poisson bracket for classical mechanics, and then uses it to do deformation quantization (related to Wigner approaches). So you consider a classical manifold and search for a non-commutative bilinear "star-operation" * on it's functional algebras so that f(X,P)*g(X,P)-g(X,P)*f(X,P) is isomorphic to the operator algebra [f, g](X,Y) you have in a Hilber space.
Apart from even-dimensional forms being defined on a manifold, there's also structures for odd-dimensional ones.
There's e.g. contact geometry
https://en.wikipedia.org/wiki/Contact_geometrywhich has been used to formulate phenomenological thermodynamics (which tends to be super awkward in the classical presentation, because there's often extensive quantities deltas "delta X:=..." defined without there being a potential "X" for them)
