We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proo
f of the Poincar{e}s theorem: a planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem ``to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3. Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable.
In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy repr
esentations of properly convex foliated projective structures on the unit tangent bundle of the surface. From this we also deduce a geometric description of the Hitchin component the symplectic group PSp(4,R).
We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equ
ation, and construct their mini--twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle $T^*N$ of any projective structure $(N, [ abla])$. We show that there is a canonical Einstein of metric on an $R^*$--bundle over $T^*N$, with a connection whose curvature is the pull--back of the natural symplectic structure from $T^*N$.