Is tangent vector on so(3) a matrix?

Question

I have read that the tangent vectors on manifolds are differential operators, looking at a particular example, the tangent space to $SO(3)$, ($SO(3)$ is the group of rotations in $\mathbb{R}^3$, which is a Lie group), on the identity matrix, is the set of antisymmetrical matrices, and is called the Lie algebra $so(3)$. If $A\in so(3)$ we can talk about a curve on $SO(3)$ given by: $$ \alpha (s)= e^{sA}, \quad \alpha(0) = e^{0} = \mathbb{I}, \quad \frac{d \alpha}{ds}\Big|_{s=0} = Ae^{sA}\Big|_{s=0} = A. $$ Therefore in this case the matrix $A$ is a tangent vector right?, my conclusion is that differential operators are linear transformations, so they have an associated matrix, the matrix A would be that associated matrix, is that right?, if isn't, how can I see the tangent vector explicitly as a differential operator in this particular case?. If having any other examples on this would be great. Thanks in advance.

Answer 1

Differential operators are linear transformations, yes, but they're linear maps $C^{\infty}(M)\to\Bbb{R}$, the domain is an infinite-dimensional vector space. They don't have a matrix-representation (which is only for maps between finite-dimensional spaces, after choosing bases on the domain and target). The point is that if you have a submanifold $S$ of a vector space $V$ (in your case $S=\text{SO}(3)$ and $V=M_{3\times 3}(\Bbb{R})$), then there is a standard linear injection $T_pS\hookrightarrow V$, so we can identify $T_pS$ with its image under this linear injection.

In the case of $S=\text{SO}(3)$ and $V=M_{3\times 3}(\Bbb{R})$, if you think of $S$ as a manifold, then the tangent space at the identity $T_IS$ is an abstractly-defined vector space, which set-theoretically is very complicated. But, after injecting it into $V$, it is just $\mathfrak{so}(3)$, the space of skew-symmetric matrices $3\times 3$ real matrices, and this is where your $A$ lives.

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I have several answers regarding similar issues:

1. Why is a vector field on the sphere equivalent to $f:S^n\to\Bbb{R}^{n+1}$ such that $f(x)\perp x$? Here I talk about how to relate the abstract definition to the more intuitive notion.
2. Calculating the derivative of a differentiable map between manifolds. Similar thing where I explain the theory, but also describe how to apply it for the specific computation for a map $f:S^2\to\Bbb{R}$.
3. How to show that $T_{(1,0)}S^1\cong \text{span}(\{e_2\})$? A similar thing, applied to the special case of the circle $S^1$.
4. What is meant by part (e) of MTW Exercise 9.13 involving the tangent to a curve on the manifold $\text{SO}(3)$? This is a PhySE answer where I explain similar stuff in the context of the manifold $\text{SO}(3)$. In that link, what I write $\Phi(T_I(SO(3)))$ is precisely equal to $\mathfrak{so}(3)$.

There may even be more answers along these lines, but this should be enough. I suggest reading them in order (though you're of course free to do it in whichever order you like).