Mathematics – Differential Geometry
Scientific paper
2011-10-09
Mathematics
Differential Geometry
30 pages. A proof of rank bound added
Scientific paper
We propose the Legendrian web in a contact three manifold for a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive relation among the first integrals of 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 show that $\, \rho_d$ is the maximum rank of a Legendrian $\, d$-web. The structure equations for the Legendrian 3-webs of maximum rank three are determined, and their explicit local normal forms are given. For an application, we give an alternative characterization of a Darboux super-integrable metric as a two dimensional Riemannian metric $\, g_+$ that 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.
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