Construction of the tangent bundle - Lec 10 - Frederic Schuller
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange askers cite it as the source of the definitions and claims they are trying to understand in differential geometry questions about bundles, tangent spaces and embeddings.
Recommended by 0 people across 0 different months, 2019 to 2022.
What it is
A 1h48m university-style lecture by Frederic Schuller from his series on the geometric anatomy of theoretical physics. It constructs the tangent bundle in differential geometry, building on earlier lectures covering bundles, tangent spaces and smooth manifolds.
What people said
Cites WillG · 2019-01 · Mathematics Stack Exchange · on “Must every manifold bundle be a fiber bundle?”
I have been learning differential geometry from this [LINK] lecture series by Frederic Schuller's. Here [LINK] he defines a bundle (of topological manifolds) as A triple, $(E, \pi, M),$ where $E$ and $M$ are topological manifolds and $\pi:E \longrightarrow M$ is a surjective and continuous function. Here [LINK] he defines a fiber bundle as A bundle $(E, \pi, M)$ such that $\forall p \in M: \text{preim}_\pi \big(\{p\}\big) \cong F$ for some manifold $F$ . Using these definitions (which I hope are standard), does there exist some bundle of topological manifolds that is not a fiber bundle (i.e., has fibers that are not homeomorphic to each other)? Sidenote, Schuller provides examples here [LINK
Cites Schatzmeister · 2022-04 · Mathematics Stack Exchange · on “Intuition behind stronger version of Whitney embedding theorem”
While watching this lecture [this video] on differential geometry, the lecturer mentions (at around 1:20:00) a stronger embedding theorem than the one by Whitney. The stronger version claims that every $d$ -dimensional manifold can be embedded into $\mathbb{R}^{2d - a(d)}$ , where $a(d)$ is the number of ones in the binary expansion of $d$ . Unfortunately, the lecturer doesn’t give any sources, so I really wonder how this (seemingly) completely unrelated notion comes up here. Does this theorem have a name? What is the intuition behind it?
Cites Kleber · 2022-10 · Mathematics Stack Exchange · on “Are tangent spaces at different points disjoint according to this definition?”
In this [this video] lecture by Prof. F. P. Schuller (I've included the correct time stamp) it is claimed that two tangent spaces at different points are disjoint, i.e. $T_pM\cap T_qM =\emptyset$ for $p,q\in M$ and $p\neq q$ . In this [LINK] lecture, the tangent space at a point $p\in M$ was defined as $$ T_pM:=\{X_{\gamma,p}|\gamma \,\mathrm{is\, a\, smooth\, curve\, through\,p}\} \quad,$$ i.e. the collection of the directional derivatives [LINK] along all (smooth) curves through $p$ : $$X_{\gamma,p}:\mathcal C^\infty(M) \longrightarrow \mathbb R$$ with $$ X_{\gamma,p} : f\mapsto \left(f\circ\gamma\right)^\prime(0) \quad ,$$ where $\gamma(0)=p$ . I don't understand the aforementioned claim. If w