A model of smooth projective plane Is there a  model of smooth projective plane $(P,L,F)$  with the  following  property?

The point space $P$  admits a Riemanian metric such that for  every point $p\in P$ and  every  line $\ell_{1}$ not  containing $p$, there is  a  line $\ell_{2}$ passing  $p$  such that $\ell_{1} \bot \ell_{2}$?

 A: Here is a partial answer and a comment:
First, the partial answer:  If the dimension of $P$ is $2n=2$, then I believe that this orthogonality property holds for any Riemannian metric that one chooses on $P$.  Here is the argument:  
Fix a Riemannian metric $g$ on $P$.  
For a given line $\lambda\in\Lambda$, let $N_\lambda\subset P\times \Lambda$ consist of the pairs $(p,\mu)\in F$ for which $(p,\lambda)\in F$ and $T_p\bar\mu$ is $g$-perpendicular to $T_p\bar\lambda$.  (Here, I am using the standard notation $\bar\lambda$ to denote the curve of points (which is a smooth embedded submanifold of dimension $1$) in $P$ that are incident with $\lambda\in\Lambda$.)  Then $N_\lambda$ is an embedded circle in $P\times\Lambda$ that projects bijectively onto $\bar\lambda\subset P$.
Let $K_\lambda\subset N_\lambda\times P\subset P\times\Lambda\times P$ denote the set of triples $(p,\mu,q)$ such that $(q,\mu)\in F$ and $(p,\mu)\in N_\lambda$.  (In other words, $(p,\mu,q)\in K_\lambda$ if and only if $\mu$ is a line incident with $q$ such that $\bar\mu$ and $\bar\lambda$ meet $g$-orthogonally at $p$.)  Then the projection $\pi(p,\mu,q)=(p,\mu)$ makes $K_\lambda$ into a circle bundle over $N_\lambda$ that is a Klein bottle topologically (and smoothly).  The map $\sigma(p,\mu)=(p,\mu,p)$ is a smooth section $\sigma:N_\lambda\to K_\lambda$. 
Now, the projection $\psi(p,\mu,q) = q$ is a smooth map $\psi:K_\lambda\to P$.  If $(p,\lambda)\in F$, then, one knows that $\psi^{-1}(p) = (p,\mu,p)$ where $\mu$ is the unique line such that $(p,\mu)\in N_\lambda$.  Moreover, $\psi:K_\lambda\to P$ is easily seen to induce a diffeomorphism from an open neighborhood of $\sigma(N_\lambda)$ in $K_\lambda$ onto an open neighborhood of $\bar\lambda$ in $P$.  It follows that $\psi:K_\lambda\to P$ is of 'odd degree'; consequently, for regular $\psi$-values $q\in P$, the cardinality of $\psi^{-1}(q)$ is odd.  (Note that this 'oddity' is the best one can do because both $P$ and $K_\lambda$ are non-orientable compact surfaces without boundary.)  In particular, $\psi$ is surjective, since, if $q\in P$ were not in the image of $\psi$, it would be a $\psi$-regular value with an even number of preimages.  Thus, for any $q\in P$ there exists a $(p,\mu,q)\in K_\lambda$.  By definition, $\bar\mu$ passes through $q$ and meets $\bar\lambda$ $g$-orthogonally (at $p$).
Second, the comment: I think this 'orthogonality property' does not hold in general when the dimension of $P$ is $2n>2$.  The problem (and the reason that the proof above does not work when $n>1$) is that it may not be possible to find a metric $g$ such that, for every $\lambda\in\Lambda$, the corresponding set $N_\lambda$ defined above is a copy of $\bar\lambda$.  The reason is that the condition that, for each $p\in \bar\lambda$, the $g$-orthogonal subspace to $T_p\bar\lambda\subset T_pP$ be the tangent $n$-plane to a $\mu\in\lambda$ such that $p$ lies on $\bar\mu$ now puts severe algebraic restrictions on the metric $g$, potentially more restrictions than can be satisfied by any metric.
To see why, consider the case $n=2$, which could be a small but 'arbitrary' deformation of the incidence relation of complex lines in $\mathbb{CP}^2$.  Then, at a point of $p$, the set of $2$-planes that are tangent to the surfaces $\bar\lambda$ for $(p,\lambda)\in F$ is a $2$-sphere $\Sigma_p$ embedded in the Grassmannian $\mathrm{Gr}_2^+(T_pP)$ of (oriented) $2$-planes in $T_pP\simeq\mathbb{R}^4$, which has dimension~$4>2$.  Such a $\Sigma_p$ has to be 'approximately' the standard 'diagonal' embedding of $\mathbb{CP}^1\simeq S^2$ into $\mathrm{Gr}_2^+(\mathbb{R}^4)\simeq S^2\times S^2$, but, as far as I can see, that's about all you can expect to know about it for an 'arbitrary' small deformation of the standard structure.  The 'generic' such $2$-sphere will not be preserved under the involution $E\to E^\perp$ induced on $\mathrm{Gr}_2^+(\mathbb{R}^4)$ by any metric $g$ on $\mathbb{R}^4$.  (After all, there is only a $9$-parameter family of such involutions.)  In fact, generically, what you would expect is that $\Sigma$ and $\Sigma^\perp$ would only meet in a finite number of points, no matter which metric $g$ you chose in order to define the perpendicular. (Even worse, when $n=4$ or $8$, you wouldn't expect $\Sigma$ and $\Sigma^\perp$ to meet at all for most metrics $g$.)
The upshot is that, one expects that, for a generic small perturbation of the standard smooth projective plane structure in dimension $2n=4$, for any metric $g$ on $P$, the set of pairs of lines that meet $g$-orthogonally will be a rather small set, too small to provide the orthogonality relations needed to satisfy this orthogonality property.
