Suppose $M\in U(n)$. Then multiplication by $M$ induces a smooth action on $U(n)/O(n)$. How can we compute the differential of this map?
If $M$ were acting on a matrix group, then of course the differential is just $M$ again, because the map is linear. But I don't know how to proceed in the presence of a quotient.
I have managed to identify the tangent space at $I$ with the set of real $n\times n$ symmetric matrices.
I'm interested in this question because I want to construct a left-invariant metric on $U(n)/O(n)$. If $A$ and $B$ are in the tangent space at zero (real-symmetric matrices), let us define
$$\langle A,B\rangle = \operatorname{tr}(AB^T).$$
If this defines a metric at $I$, we can just translate it everywhere and be done. But the source I'm using notes we must verify this metric is invariant under the action of $O(n)$. The action is supposedly
$$g\rightarrow gAg^{-1}$$
on the tangent space. Why is that? I'm a little confused about the set-up here.