I’m reading Walter Thirring’s treatment of tangent spaces in Classical Mathematical Physics and have been struggling with the meaning of some of the abstract notation. Specifically, I don’t understand his treatment of charts and atlases for tangent bundles.
I hoped I would be helped by one of his concrete examples—stereographic projection—but unfortunately all my attempts to understand it have failed.
In particular, I don’t understand where $\Theta_{C_\pm}$ in this image comes from:
The abstract definitions appear as follows. In general:
And in the context of a tangent bundle:
As you can see, Thirring refers to the first definition, $\Theta_C(q)$, as a “mapping”, which is so generic that it makes it impossible to search for, and other treatments of this subject (of which I have now read many) don’t connect to Thirring’s discussion in any obvious way.
Any help would be much appreciated.
It seems like what he's referring to two things:
The first is that vectors in $T_q M$ can be described using an equivalence class of curves that pass through $q$ such that their images under the chart all have the same velocity (tangent) vector at $\Phi(q)$. So when trying to specify a vector in $T_q M$ one can simply provide a curve from the equivalence class.
The second would be what is called a local trivialization. Which is to say that locally the tangent bundle looks $\Phi(U)\times \mathbb{R}^n$. In other words there is a (smooth) map $$\Theta_C: TU\rightarrow \Phi(U)\times \mathbb{R}^n$$ That satisfies $\text{pr}_1\circ \Theta_C(q,v)=\Phi(q)$ where $\text{pr}_1$ is the projection onto the first factor. This condition ensures that $$\Theta_C(q,v)=(\Phi(q), w)$$ for some $w\in \mathbb{R}^n$. The map $\Theta_C (q)$ is precisely the map which takes $v$ and assigns $w$. I.e. the unique map satisfying $$\Theta_C(q,v)= \left(\Phi(q), \Theta_C(q)\cdot \left( v\right)\right)$$
So in the given problem, he specifies how a point $(\mathbf{x},\mathbf{v})$ in a local neighborhood maps under the chart by first identifying the vector as a curve passing through $\mathbf{x}$, and secondly specifying the local trivialization $\Theta_{C_\pm}$ which has the chart in the first component:
$$\Phi_\pm(x_1,x_2,x_3)= \frac{(x_1,x_2)}{1\mp x_3}$$
And how the vector "pushes forward" with the $\Theta_{C_\pm}(x_1,x_2,x_3)$ in the part after the semicolon.
For the specific given example the idea for the chart is the following: consider any point $\mathbf{p}$ in $S^2\setminus \{\mathbf{n}\}$ where $\mathbf{n}$ is the "north pole" of the sphere, there is a unique point in the plane corresponding to that point on the sphere. The way to see this is if you consider the sphere to be embedded in three dimensional space, you can draw a line from the north pole to $\mathbf{p}$ and the line will intersect the plane at a unique point. The map which assigns $\mathbf{p}$ to its corresponding point in the plane is the chart $\Phi$.
Constructing the equation of the line is quite simple and to figure out the corresponding point in the plane, find where the line intersects the plane. This gives the first part of $\Theta_C$. The second part is using this chart to compute $D(\Phi\circ u )(0)$ where $u$ is any curve from the equivalence class of a vector in $T_\mathbf{p}S^2$.