The tangent space to $SO(3)$ at the identity can be identified as the space of all skew symmetric matrices $Skew_3$.
At other points $X$, we can see that the tangent space is the set of all matrices $Y$ such that $Y X^t + X Y^t = 0$, or $(YX^t) = -(YX^t)^t$.
- Are these known by any specific name?
- Why aren't these set of matrices $Y$ studied, except when $X=\mathbb{I}$?
I wonder if we can impose additional (useful) structure, such as an inner product (and therefore a norm) on $Skew_3$ so that we can reason about its "shape."
- Is it flat, like the Euclidean space $\mathbb{R}^3$ or is it a curved space? The reason I ask is because tangents to curves/spaces embedded in $\mathbb{R}^2$ (e.g., circle) or $\mathbb{R}^3$ (e.g., sphere) look flat (line and plane respectively). I am curious whether this intuition that tangent spaces are "flat" translates to other manifolds.
1) The tangent space at each point is a copy of $\mathfrak{so}_3$ (aka $Skew_3$). As $SO(3)$ is a Lie group all of its tangent spaces are isomorphic and carry the structure of a Lie algebra.
2) I'm not 100% sure on this but I think the relation you have described is simply created by the action of conjugation of the element $X \in SO(3)$ on the relation $Y^t = - Y$ and so studying it is equivalent to studying $\mathfrak{so}_3$ as the set of skew-symmetric matrices (remember that defining it with matrices already involves a choice of basis and this would be equivalent to that)
3) All tangent spaces are vector bundles. That is at each point the tangent space is a vector space which is a fundamentally flat object.