Rigorous computation of $T_xS^1$

Question

I'm reading Guillemin and Pollacks book on Differential Topology, and in there, there's the following exercise:

Calculate $T_xS^1$ where $x=(a,b) \in S^1 \subset R^2$ in terms of $a$ and $b$. (Of course the answer is the line spanned by $(-b,a)$, but prove it).

It's not clear to me what is expected of me. The definitions are:

Given a smooth map $f$ between open sets in $\mathbb{R}^n$ and $\mathbb{R}^k$, define:

$$df_x(h) = \lim_{t \to 0} \frac{f(x+th) - f(x)}{t}$$

which is a linear function $\mathbb{R}^n \to \mathbb {R}^k$

Given a manifold $X \subset \mathbb{R^N}$, and a local parametrisation $\phi:U \to X$ around $x \in X$, with $U$ open in $\mathbb{R}^k$, and $\phi(0) = x$, we put:

$$T_xX = \text{Im } d\phi_0 \subset \mathbb{R}^N$$

Now: In the case of the circle, given any $(a,b)$, I first need a local parametrisation $\varphi:\mathbb{R} \to S^1$ around $(a,b)$, that must satisfy the that $\varphi(0) = (a,b)$. I am supposed to calculate this $\varphi$ explicitely? I could use the four standard charts on $S^1$, i.e. mapping the four semi-cirle to the real line, but then $\varphi^i(0) = (a,b)$ doesn't hold for an arbitrary $(a,b)$. So for any $(a,b)$, I would have to calculate different open sets in $S^1$, and get different parametrisations of these different open sets.

And after all that I need to compute the image of $d\phi_0$.

Can you see that this is very confusing?

I have seen arguments relying on euclidean geometric facts, i.e. this line in $\mathbb{R^2}$ touches the circle at this one point, its perpendicular to the normal of the circle at this point, and here is that lines equation. But this is something very different than showing that the tangent space is the image of a map, and has nothing to do with GPs theoretical exposition.

So what is expected of me to do in this situation? And I can't find a single worked example, I'm expecting something like "In this situation, we have":

$U_x = ...$,

$\phi_0 = ...$,

$T_xX = \text{Im } \phi = \text{...[calculations...]} =\text{Vectors spanning $T_xX$}$

So what does G&P expect I do? What steps do I take to make a calculation that fits with the theory?

Answer 1

Yes, you need to find a parametrization around for $S^1$ around $x$ but we can do it by modifying the "obvious" parametrization. Assume that $x = (a,b) \in S^1$ and $x \neq (1,0)$. Let $\psi \colon (0,2\pi) \rightarrow X$ be the parametrization $$\psi(\theta) := (\cos(\theta), \sin(\theta)). $$

Since $x \neq (1,0)$, we know that we can find a unique $\theta_0 \in (0,2\pi)$ such that $\psi(\theta_0) = x$. We want to construct another parametrization $\phi$ of $S^1$ such that $\phi(0) = x$. We can do it by repearametrizing $\psi$ as follows:

$$ \phi(\theta) := \psi(\theta + \theta_0) = (\cos(\theta + \theta_0), \sin(\theta + \theta_0)). $$

Then $\phi \colon (-\theta_0, 2\pi - \theta_0) \rightarrow S^1$ is a parametrization of $S^1$ that satisfies $\phi(0) = x$ and so

$$ T_{x}(S^1) = \operatorname{im} d(\phi)|_{0} = \operatorname{im} ((-\sin(\theta_0), \cos(\theta_0))) = \operatorname{span} \{ (-\sin(\theta_0), \cos(\theta_0))\} = \operatorname{span} \{ (-b, a) \}.$$

I'll leave the case $x = (1,0)$ to you.