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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.

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    $\begingroup$ 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. $\endgroup$
    – Didier
    Oct 4, 2021 at 14:42
  • $\begingroup$ @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? $\endgroup$ Oct 4, 2021 at 15:57
  • $\begingroup$ 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. $\endgroup$
    – Didier
    Oct 4, 2021 at 16:11
  • $\begingroup$ 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 $\mathbb{RP}^{2}$ $\endgroup$ Oct 4, 2021 at 16:22

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