Define as usual $\mathbb{RP}^2:=(\mathbb{R}^3\setminus \{0\})/\mathbb{R}^*$ and on $\mathbb{RP}^2\setminus \{X=0\}$ consider this riemannian metric $g=\frac{dX^2+dY^2+dZ^2}{X^2}$ (where $X,Y,Z$ are homogeneous coordinates).
Question: how is $g$ written in homogeneous coordinates?
Hwang explained my how to write in homogeneous coordinates the round metric $g_r=\frac{dX^2+dY^2+dZ^2}{X^2+Y^2+Z^2}$ on the whole $\mathbb{RP}^2$, but I can't understand how to apply the reasoning on $g_r$ to $g$. In particular is the decomposition in radial and tangential vector done in the same way? What should I subtract to $\frac{dX^2+dY^2+dZ^2}{X^2}$?
Thank you!
$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Proj}{\mathbf{P}}$Hint: If $\pi:\Reals^{3} \setminus\{(0, 0,0)\} \to \Reals\Proj^{2}$ denotes the projection and if $g$ is the unit round metric on the projective plane, then $\pi^{*}g$ is $O(3)$-invariant, scale-invariant, restricts to the round metric on the unit sphere, and is degenerate along lines through the origin. The symmetric two-tensor $$ \frac{dX^{2} + dY^{2} + dZ^{2}}{X^{2} + Y^{2} + Z^{2}} $$ (which is probably what you were trying to write down) is $O(3)$-invariant, scale-invariant, and restricts to the round metric on the unit sphere, but has a non-zero radial component that must be subtracted off: \begin{align*} \pi^{*}g &= \frac{dX^{2} + dY^{2} + dZ^{2}}{X^{2} + Y^{2} + Z^{2}} - \frac{(X\, dX + Y\, dY + Z\, dZ)^{2}}{(X^{2} + Y^{2} + Z^{2})^{2}} \\ &= \frac{(X^{2} + Y^{2} + Z^{2})(dX^{2} + dY^{2} + dZ^{2}) - (X\, dX + Y\, dY + Z\, dZ)^{2}}{(X^{2} + Y^{2} + Z^{2})^{2}}. \end{align*}