Definition talk:Tangent Space

Missing definition?

I cannot find any relation to Definition:Tangent Vector. One of the definitions of the tangent space $T_p M$ should be the set of tangential vectors at $p$. Equivalence of Definitions of Tangent Space should show the equivalence of such and the one given in this page. In particular, the current redirection seems inappropriate. Is this perspective completely missing in $\mathsf{Pr} \infty \mathsf{fWiki}$? --Usagiop (talk) 00:37, 9 November 2022 (UTC)

The problem is that we have never covered a single textbook on the manifold theory to its full extent. Obviously one should deal first with the topological manifolds, then differentiable manifolds, then Riemannian manifolds. However, I am only working on the third case because this is what I need for my literature review. Once I am done with that, I would love to start from the basics and build up all this stuff properly. For now, anyone is welcome to fill in any gaps present.--Julius (talk) 01:50, 9 November 2022 (UTC)

My point was only that the relation between Definition:Tangent Space and Definition:Tangent Vector seems completely missing. However, now, I have seen there were actually some I overlooked. The section Differentiable Manifold and 3 different definitions under it must be separated as Definition:Tangent Space/Differentiable Manifold, Definition:Tangent Space/Differentiable Manifold/Definition 1, ... --Usagiop (talk) 19:56, 9 November 2022 (UTC)

Indeed we should refactor this page for the definitions to have their own pages. I think Definition:Tangent Space and Definition:Tangent Vector at least share the notion of tangent vectors as derivations, so there is some connection between both pages. But the tangent space at a point can also be understood as a set of tangent vectors of all possible curves passing through that point. I don't think we have this notion. Really, at the moment both pages have not been properly harmonized.--Julius (talk) 23:23, 9 November 2022 (UTC)

Yes, this is exactly what I mean. Thanks for the clarification. --Usagiop (talk) 00:16, 10 November 2022 (UTC)

Contributors are strongly discouraged from posting stuff up that they have not sourced either from a hard copy or from a select number of trusted sites. This is in reaction to a number of cases where an editor has posted up unsourced stuff which turned out to be incorrect. It is a sad fact that such contributors can sometimes be less than receptive to the idea that they may not be quite as expert on a subject as they believe themselves to be, and there has been a certain amount of friction in the past.

Hence our exhortation to generate a body of knowledge on a particular field from the ground up, starting from the front of a book and working through to the back. Again, we have had contributors jump straight into the middle of a complex work, believing that the material in the early part of the book is "already covered", not realising that the approach may have been significantly different from that taken by other source works, resulting in compromised accuracy. --prime mover (talk) 18:28, 9 November 2022 (UTC)

I would like to define a manifold using pseudogroups since it gives a very neat characterisation that will let us prove some very general theorems. Don't know a source for this off the top of my head, I had it lectured to me in a DG course, but it shouldn't be too hard to find one. Caliburn (talk) 19:05, 9 November 2022 (UTC)