Yup. Cartan understood it :) You think of it as analogous to curvature for the turning of a vector but now for the base point of the tangent vector (affine connection).
But torsion in the Riemannian case makes sense only for the tangent bundle, not general bundles. There are different notions of torsion, e.g., for projective connections.
@AnindyaPrithvi I would request you to write your question in a normal font size rather than what you're doing. This just discourages even more people from trying to help you, and looks horrible.
Hi all, quick question. Let's say I have a union of two intervals $(a, c] \cup [d, b) \subset \mathbb{R}$. How can I show that it cannot be written as a finite union of open intervals? I strongly suspect that this has to do with some facts from topology that I'm not aware of.
If $X: M\rightarrow TM$ is a smooth vector field such that $X_p=0$ and $f: \mathfrak{X}(M)\rightarrow C^{\infty}(M,\mathbb{R})$ is a $C^{\infty}(M,\mathbb{R})$ linear map, does it follow that $f(X)(p)=0$?
@Astyx it goes as product of $(1+1^-)(1+1^{--})....(1+1/2)$ as I reach n...also, a mistake in writing the question, I meant the whole expression to the power of 1/n
@TedShifrin My topology background is extremely weak (I never actually took a class on it). So if I'm getting what you're saying, with $\mathbb{R}$ under the standard topology, any union of open intervals is open, and what we have here is half-closed intervals (which aren't open), so I can't represent the union above as a disjoint union of open intervals, done.
I hope I don't regret taking measure theory this semester. It seems like the prof expects us to know quite a bit of topology coming in. I'm a bit annoyed at the fact that my analysis classes in $\mathbb{R}$ and $\mathbb{R}^n$ did not cover topology.
note that i'm trying to show that if we have a $f: \mathfrak{X}(M) \rightarrow C^{\infty}(M) C^{\infty}(M)$ map, then there exists a smooth one form $g$ such that $g(X)=f(X)$ for all smooth vector fields X
Right, @orientable. There is a basic lemma that you should have seen (I presume it's in Lee) that characterizes tensors as such maps that are linear over $C^\infty$. It's in every differentiable manifolds course/book.
@Clarinet: You don't need to be so fancy as to do stuff with Munkres, although having seen a basis for a topology, for example, will be helpful in measure theory.
Did stirling use the fact that if an integer is to be evenly broken such that the product is maximum.....the broken piece should have size [n/e] where [] is G.I.F.
@TedShifrin Perhaps I'm oversimplifying it, but to me, a topological space is just another type of collection of subsets of a set like you see in measure theory with rings, semirings, algebras, $\sigma$-algebras, etc. But yeah, I'll definitely be getting through the basics. I'm going to at least learn what the terms are.
My list includes: topological space, open, closed, connectedness, compactness, basis for a topology, interior, closure, subspace/product/quotient topology, maybe some things on metric spaces.
I -think- that will suffice
I'm glad I did some studying already over the summer, since I hadn't learned about the subspace topology and was confused when one of the texts I was using to study used it without explaining
@Clarinetist another great topology textbook is Mendelson's book. It's a Dover book, so it's like 15 USD.
It's also short. It starts with motivation from metric spaces and then generalizes to topology. It's a very nice book and highly recommend it to anyone wanting to learn topology in a smooth, but succinct manner.
Mind you if you have other texts, you might as well try learning from those, but if you have the money to spare and don' mind Yet Another Topology Textbook, then I guess why not.
Thanks all for your help! I'll leave you with a nice tidbit that I learned since when I was last here
This is one of the most ambitious books I've ever seen, titled "Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning:" cis.upenn.edu/~jean/gbooks/geomath.html
Past the point of knowing the very basics (defn of topology, continuity, metric topologies, product topologies, subspace topologies, bases, and maybe closure/interior) it starts to get to the point where you can learn whatever you want. The standard topics after that are connectedness, compactness, Hausdorff spaces, and quotient spaces, which I would expect to be covered in any intro course.
After that you get into more specialized territory
I like Hatcher's notes precisely because they're so short and cover nothing more than they need to.
Here there's a certain number of classes you're supposed to take, basically I'm taking 3 per term this year, took only 2 per term last year, and third and fourth year 2 per term
Mostly math though next year I might try to do some ML and/or stats for the sake of getting jobs lol
If $\{M_i \mid i \in I\}$ is a collection of modules, how is $\bigoplus_{i \in I} M_i$ defined? Does it consist of all tuples with at most finitely many nonzero entries?
