The Geometry of the Projective Action of $\text{SL}(3,\mathbb{R})$ from the Erlangen Perspective

In this paper, we have investigated the projective action of the Lie group $\text{SL}(3,\mathbb{R})$ on the homogeneous space $\mathbb{RP}^2$. In particular, we have studied the action of the subgroups of  $\text{SL}(3,\mathbb{R})$ on the non-degenerate conics in the space $\mathbb{RP}^2$. Using the Iwasawa decomposition of $\text{SL}(2,\mathbb{R})$, we demonstrate that the isotropy subgroup of the projective unit circle is isomorphic to  $\text{PSL}(2,\mathbb{R})$ under certain conditions.