We define a relation in the sphere by identifying the antipodal points, the quotient space obtained is the projective plane $\mathbb{P}^2$. Also, the quotient map $\pi:\mathbb{S}^2\longrightarrow\mathbb{P}^2$ turns out to be a double covering of the projective plane. Given an isometry $T$ in $\mathbb{P}^2$ we have a lifting $\widetilde{T}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$ such as $T\circ\pi=\pi\circ\widetilde{T}$.
My first question is: is this lifting $\widetilde{T}$ unique? And if it is not, can we make it unique? How? Well, that have been three questions.
My second question is: Can we use the previous to show that $\widetilde{T}$ is an isometry on the sphere? How?
My third question is: If we show that $\widetilde{T}$ is an isometry, how this help us to get all the isometries in $\mathbb{P}^2$?
The next part of the problem is about the geodesics in $\mathbb{P}^2$, Any hint about it? Maybe using the quotient map $\pi$?
