I am trying teach myself the basics of curvature by doing concrete computations. I would like to check the Gauss-Bonnet formula for the real Grassmannian $G(1,3)$ of lines in $\mathbb{R}^3$, which is nothing but the real projective plane. I see it as the set of all rank 1 projections in $M_3(\mathbb{R})$, which sit in the unit sphere of $\mathbb{R}^9$ when we equip $M_3(\mathbb{R})$ with the Frobenius norm. If possible, I would like to use the following parametrization $$(s,t)\in\mathbb{R}^3\longmapsto\frac{1}{1+s^2+t^2}\begin{bmatrix}1&s&t\\s&s^2&st\\t&st&t^2\end{bmatrix}$$ for all but one element of $G(1,3)$.
Question 1: How does one compute the Gaussian curvature $\mathcal{K}$ at a given point using the latter?
Question 2: What is the area form $\mathrm{d}A$ at such a point in the formula and how can I integrate the Gaussian curvature and check the Gauss-Bonnet formula $$\int_{G(1,3)}\mathcal{K}\,\mathrm{d}A=2\pi\chi(G(1,3)) \,?$$
Remarks:
- I already know how to show that $\chi(G(1,3))=1$, so this question is really about the LHS of the formula above.
- Since this parametrization covers the whole $G(1,3)$ but one point, I expect it is enough to use the above parametrization, although I'm note sure.