# Does the tangent space to $SO(3)$ have additional useful structure?

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 $$(YX^t) = -(YX^t)^t$$.

1. Are these known by any specific name?
2. 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."

1. 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.
• math.stackexchange.com/questions/1784898/… Commented Oct 2, 2018 at 16:35
• math.stackexchange.com/questions/908137/… Commented Oct 2, 2018 at 16:36
• Interesting, thanks for the links @cactus314! The links answer my intuitions about (3). Do you have any pointers for (1) or (2)? Commented Oct 2, 2018 at 17:09
• ad 1), that's the Lie algebra $\mathfrak{so}_3$ I guess, but since you already put the tag "lie-algebras" in there, I would assume you know that, and all the nice things one can find about a Lie group via its Lie algebra? ad 2), if I recall correctly, the tangent spaces at different points of a Lie group are all isomorphic, using the group structure (isomorphisms from the space at $1$ to the space at an arbitrary $g$ induced by either left or right multiplication with $g$). Commented Oct 3, 2018 at 21:56
• Thanks @TorstenSchoeneberg. For (2), it didn't occur to me that tangent spaces at different points are isomorphic to each other (it makes sense since $X$ is a rotation, so it's invertible by $X^t$). For (1), while I know it has a Lie Algebra structure, but I wondered if it can be endowed with any other structure (such as a suitably defined inner product) that is useful in practice. Commented Oct 3, 2018 at 22:33

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)