What is the tangent space o SO(n)

Question

It should be the kernel of the map $H\mapsto A^TH+H^TA$ at some $A$ such that $A^TA=I$ . But I cant find this Kernel can someone help me?

Answer 1

The tangent space at $I_n$ is the space of antisymmetric matrices defined by $A+A^T=0$. Given $g\in SO(n)$, and let $AS(n)$ the space of antisymmetric matrices, the tangent space at $g$ is $gAS(n)$. That is the image of the tangent space $T_{I_n}SO(n)$ by the left translation defined by $g$.

Suppose that $g\in SO(n)$, $A\in AS(n)$, you have $$g^T(gA)+(gA)^Tg=(g^Tg)A+A^Tg^Tg=A+A^T=0.$$