If $SL(2)=\{A\in \mathcal{M}_2: \det(A)=1\}$, find a parametrization around the identity matrix $I_2$ and find the first fundamental form.
I've proved that $SL(2)$ is an hypersurface in $\mathbb{R}^4$, so it has dimension $3$. However, I don't understand very well what does "around the identity" means. I think that if I have the parametrization I'll be able to compute the first fundamental form.
Thank you.
On
Since $\mathrm{SL}_2$ is a quadric hypersurface, one way to get a parametrization near the point $I_2$ is to stereographically project $\mathrm{SL}_2$ onto the tangent hyperplane at $I_2$ from some other point (for instance, $-I_2$).
The group is the set of matrices $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ such that $ad-bc=1$. If you fix $a$, $b$ and $c$, then you can solve for $d$ using a somewhat obvious formula, and that formula works in a neighborhood of the point $(a,b,c)=(1,0,0)$ which gives the identity. This gives you a parametrization of the many points on the group including the identity. I'll let you write it down explicitly and seeing if it gives a chart or not.