Invariant Projective Properties Under the Action of the Lie Group \(\textrm{SL}(3;\mathbb {R})\) on \(\mathbb{R}\mathbb{P}^2\)

Abstract

In this paper we define the projective action of the Lie group \(\textrm{SL}(3;\mathbb {R})\) on \(\mathbb{R}\mathbb{P}^2\). We have considered all the one-parameter subgroups (up to conjugacy) of \(\textrm{SL}(3;\mathbb {R})\) and constructed their orbits in two-dimensional homogeneous space by defining the projective action. We obtain the underlying geometry under this action of \(\textrm{SL}(3;\mathbb {R})\) by finding the corresponding invariant projective properties. We also discuss whether the action of \(\textrm{SL}(3;\mathbb {R})\) is triply transitive and to find the possible fixed points under the action.