The real projective plane $\mathbb RP^2$ is introduced in algebraic geometry contexts as basically $\mathbb R^2$ but where all lines intersect at one point. One construction is the one that projects the 3D-world onto our 2D field of vision (the original starting point of the field?), by having an "eye" $E$ at $(0,0,1)$ looking down at the $xy$-plane, mapping each point $(x,y,0)$ to the intersection of the [line between $E$ and $(x,y,0)$] and the [(lower hemisphere of the) sphere $S$ centered at $(0,0,1)$], adding in "points at $\infty$" on the equator of $S$ (identifying antipodal points on that circle).
Lines passing through $(0,0,0)$ in $\mathbb R^2$ are obviously mapped to (half of a) great circles on $S$, and I'm pretty sure other lines in the $xy$-plane are also mapped to great circles on $S$ as well. This is nice, as great circles are geodesics on $S$, matching the notion of line in $\mathbb R^2$.
The bad thing is that this "model" of $\mathbb RP^2$ is not a continuous image of $\mathbb RP^2$ in $\mathbb R^3$. There are immersions of $\mathbb RP^2$ (I in fact asked about it recently); pictures of very nice ones can be found here: http://wordpress.discretization.de/ddg2019/2019/05/06/tutorial-4-boys-surface/. See this animated video about it showing what it's like to move around on it. A different animated construction can be found here.
Question: is there an immersion of $\mathbb RP^2$ into $\mathbb R^3$ so that lines (the great circles in the hemisphere model I described above) are geodesics on the immersed surface?