Maximum rank of a Legendrian web

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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 of ρd = (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 ρd is the maximum rank of a Legendrian d-web. In the complex analytic category, we give a possible projective geometric interpretation of ρd as an analogue of Castelnuovo bound for degree 2d surfaces in the 3-quadric Q³⁴ via the duality between P³ and Q³ associated with the rank two complex simple Lie group 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+ which admits a mate metric g- such that a Legendrian 3-web naturally associated with the geodesic foliations of the pair g± has maximum rank.