Maximum rank of a Legendrian web

Abstract

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 $$ \rho_d = \frac{(d-1)(d-2)(2d+3)}{6}$$ linearly independent Abelian relations. We then employ the method of local differential analysis and the theory of linear differential systems to show that $\, \rho_d$ is the maximum rank of a Legendrian $\, d$-web. In the complex analytic category, we give a possible projective geometric interpretation of $\rho_d$ as an analogue of Castelnuovo bound for degree 2d surfaces in the 3-quadric $\mathbb{Q}^3\subset\mathbb{P}^4$ via the duality between $\mathbb{P}^3$ and $\mathbb{Q}^3$ associated with the rank two complex simple Lie group Sp$(2,\mathbb{C})$. 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 $\, g_{\pm}$ has maximum rank.