# Understanding $T_{p}\mathbb{RP}^{2}$ ,tangent planes of his opens sets and transverse

I have to find two submanifolds $$N_{1}$$ and $$N_{2}$$ of dimension 1 of $$\mathbb{RP}^{2}$$ the real projective space such that $$N_{1}\cap N_{2}\neq \emptyset$$ and $$N_{1}\pitchfork N_{2}$$ .

For this I was thinking in some maps $$f_{1}$$ and $$f_{2}$$ from $$\mathbb{RP}^{2}$$ to $$\mathbb{R}$$ such that $$f_{1}$$ and $$f_{2}$$ are submersion and I got that $$f_{1}^{-1}({q_{1}})$$ and $$f_{2}^{-1}(q_{2})$$ are submanifolds of $$\mathbb{RP}^{2}$$ of dimension 1 for $$q_{1}, q_{2}\in \mathbb{R}$$, but for check that submersion I have to check that differentials $$df_{1q}:T_{q}\mathbb{RP}^{2} \rightarrow T_{f(q)}\mathbb{R}$$ and $$df_{2q}:T_{q}\mathbb{RP}^{2} \rightarrow T_{f(q)}\mathbb{R}$$ are subjective, but How I undertand $$T_{q}\mathbb{RP}^{2}$$? I know that it's homeomorphic to $$\mathbb{R}^{2}$$ so Can I understand the basis of $$T_{q}\mathbb{RP}^{2}$$ in terms of basis in $$\mathbb{R}^{2}$$ by the homeomorphism? and if I can get that two maps $$f_{1}$$ and $$f_{2}$$ in particular are defined in some opens $$U_{1}, U_{2} \subset \mathbb{RP}^{2}$$ and their are my submanifolds, but when I will try to check that $$U_{1}\pitchfork U_{2}$$ How can I understand $$T_{p}U_{1}$$ and $$T_{p}U_{2}$$? If you can help me with these question, I will gratefully. Thank you.

• One easier way: consider the projection $\pi\colon \mathbb{S}^2 \to \mathbb{RP}^2$, which is a local diffeomorphism, and two easy transerve submanifolds of the sphere. Oct 4, 2021 at 14:42
• @Didier the submanifolds on $\mathbb{S}^{2}$ are any curves on $\mathbb{S}^{2}$ or Can I consider some geodesics? and If I found this submanifolds on $\mathbb{S}^{2}$, Have I this result on $\mathbb{RP}^{2}$ about $\pi$? Maybe I have this on opens, right? Oct 4, 2021 at 15:57
• Well, if you "have to find two transverse submanifolds", you have the choice to construct them by any mean you want. Two transverse great circles on the sphere will do the job. Oct 4, 2021 at 16:11
• ohhh.. I believe these great circles are on the planes $xy$ and $xz$, this intersection is non empty and the spam of the tangents planes are ortogonal of dimension 1 respectively, so $T_{p}S_{1}+T_{p}S_{2} = T_{p}\mathbb{S}^{2}$, hence, like $\pi$ is a local diffeomorfism I got that $d\pi_{p}$ is local injective and I can traslate this result on $\mathbb{RP}^{2}$ Oct 4, 2021 at 16:22