Best products from r/dgatp

So this is a very common place to get stuck when learning about tensors. I don't know if he mentioned it in the video, but the star operation is called taking the dual of a vector space. He skips some detail in his answer, but that's understandable given the situation.

The double dual of a finite dimensional vector space is not "exactly" the same as the vector space. Its is isomorphic (a subtle distinction which sometimes helps and sometimes doesn't) to it and there is only one isomorphism that is "sensible". This is why he says the word "canonical". Wikipedia does an ok job of explaining the dual and the construction of the isomorphism

Let V be a vector space and let e_i, i=1,...,n be a basis. Define e^(i):V \to \mathbb{R} to be the linear function that takes e_i to 1 and all other basis vectors to 0. The set of functions e^(i): i=1,\ldots,n can be used to generate a vector space. This vector space is the same as the dual, V^* the space of linear functions from V to \mathbb{R}. In particular it makes sense to write things like e^(i)(e_j) and the various linear combinations of them.

Now repeat this construction define g_i to be the linear function that takes the function e^(i) to 1 and all other basis functions to 0. It now makes sense to write things like g_i(e^(j)) and the various linear combinations of them. The space generated by the g_i 's is a finite dimensional vector space of the same dimension as V so the two spaces are isomorphic. Usually there is no "sensible" choice of isomorphism. You pick a linear mapping between the basis vectors and stick with it, but in this case we have a "sensible" choice. The isomorphism will take g_i to e_i . So now we rewrite g_i(e^(i)) as e_i(e^(i)) and consider the vectors in V as functions on the 'vectors' in V .

What the notation e^(i)(e_j) and e_i(e^(j)) represents is already familiar to you. It's like the product of a row and column vector.

I've picked the upper and lower case indices important as there are conventions regarding their use in DG. Fair warning: as some point you'll see index notation and something called abstract index notation both of which use indices in a similar way but to represent different objects. It's a big source of confusion for people picking the subject up. There is a mathexchange question that does a good job of explaining the difference and pointing to further resources.

He (in the mathexchange post) doesn't make it clear, but I'm pretty sure that the Penrose book that he's referring to is by Penrose and Rindler. They spend an excessive amount of time discussing different notational devices in DG. Which I thought was worth reading once (plus you'll get a bonus weird graphical tensor notation that Penrose championed but never caught on).

Edit: because apparently Reddit doesn't use a proper latex implementation.
Edit: OMG I can't believe a site like reddit hasn't got mathjax and markdown working at the same time. Yes it's not easy, but it's also not difficult.

I could be interested in reading that paper, however I might need a discussion on the Atiyah-Singer Index Theorem first - It's something I haven't really had to use, but something I'd like to know.

My own personal interests lie in manifolds with special holonomy, and I'd be particularly interested in discussing G2 manifolds, if anyone else is.

Another, more basic, option would be Frenkel's 'Geometry of Physics' book, which has a lot of nice physics formulated in the language of differential geometry. This may be a good option for people with physics backgrounds with little formal DG training, as it does all of DG from scratch while being sure to tie all the math to physics (E&M, Lagrangian/Hamiltonian Mechanics, Relativity, Yang-Mills Theory etc.) Check it out here: https://www.amazon.com/Geometry-Physics-Introduction-Theodore-Frankel/dp/1107602602

I'm currently working through Calculus to cohomology - Madsen, Tornhave. The first half is basically setting up de Rham cohomology on manifolds (differential forms, Mayer-Vietoris, integration on manifolds). The second half contains stuff on vector bundles, characteristic classes, curvature, connections.

The material covered is great in my opinion, and it's very compact (which is a good thing or a bad thing depending on your point of view). The downside is that there's often little (or no) motivation for why you are doing certain things. It leaves the reader to fill in the motivation for themselves. I personally quite like that cause it forces me to think harder about it rather than just skating over the details.

There's two books by Spivak: Calculus on Manifolds, and A comprehensive introduction to differential geometry. The former introduces differential forms and was the text for my multivariable analysis class. I wasn't a big fan to be honest. The latter is a sequel I suppose, and is much better. Covers lots of useful stuff. Good as a reference, but personally I don't like the way Spivak writes that much (though it seems I'm the only one).

I also quite like Lee - Smooth manifolds. Covers slightly different stuff to Madsen, though a lot of overlap. Approach is quite different, more motivation is provided which is nice. I'm reading a bit of both right now.

is it that one? https://www.amazon.com/Differential-Geometry-Connections-Curvature-Mathematics/dp/0199605874

how close is this to the graduate level? because sometimes i forgot definitions and would need to look them up. does provide basics sometimes?

not a big deal with the exercises, i learned to make my own.
thanks!
maybe we can add this to sidebar!

Here is the mobile version of your link