Maximum rank of a Legendrian web

We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its foliations. For a class of Legendrian d-webs defined by simple second order ODE's, we give an algebraic construction ofρd=(d−1)(d−2)(2d+3)6 linearly independent Abelian relations. We then employ the method of local differential analysis and the theory of linear differential systems to show that ρdρd is the maximum rank of a Legendrian d-web.In the complex analytic category, we give a possible projective geometric interpretation of ρdρd as an analogue of Castelnuovo bound for degree 2d   surfaces in the 3-quadric Q3⊂P4Q3⊂P4 via the duality between P3P3 and Q3Q3 associated with the rank two complex simple Lie group Sp(2,C)Sp(2,C).The Legendrian 3-webs of maximum rank three are analytically characterized, and their explicit local normal forms are found. For an application, we give an alternative characterization of a Darboux super-integrable metric as a two dimensional Riemannian metric g+g+ which admits a mate metric g−g− such that a Legendrian 3-web naturally associated with the geodesic foliations of the pair g±g± has maximum rank.