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.