Homogeneous Integrable Legendrian Contact Structures in Dimension Five

Abstract

We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using the techniques of parabolic differential geometry, we compute the associated regular, normal Cartan connection and give explicit formulas for the harmonic part of the curvature. The PDE system is trivializable by means of point transformations if and only if the harmonic curvature vanishes identically.In dimension five, the harmonic curvature takes the form of a binary quartic field, so there is a Petrov classification based on its root type. We give a complete local classification of all fivedimensional integrable Legendrian contact structures whose symmetry algebra is transitive on the manifold and has at least one-dimensional isotropy algebra at any point.Regarding c as a function of a, b and treating x, y, u as parameters, we have c a = y + 2b(x + a), c b = (x + a) 2 , and c aa = 2b, c ab = 2(x + a) = ±2 √ c b , c bb = 0.WLOG, the ± ambiguity can be eliminated: the corresponding PDE systems are equivalent via the point transformation (a, b, c) → (−a, b, c). Thus, the dual system to III.6-1 is u 11 = 2y, u 12 = 2 √ q, u 22 = 0.Our classification indicates that III.6-1 is not self-dual (but a priori this is not at all obvious).

Exceptionally simple PDE

“…pM 2n`1 , Cq with a decomposition C " E ' F into complementary Legendrian subspaces. When F is integrable, we can introduce coordinates px i , u, u i q, 1 ď i ď n, so that C " annpdu´u i dx i q and F " spantB u k u (see [9]). Now E specifies a section of M p1q Ñ M , so such structures can be regarded as complete systems of 2nd order PDE u ij " f ij px k , u, u k q up to point transformations, i.e.…”

“…‚ n " 1: Relative invariants I 1 , I 2 (Tresse invariants). ‚ n ě 2: Two torsions τ E , τ F (obstructing the integrability of E, F ) and a curvature W. For semiintegrable structures, u ij " f ij px k , u, u k q, W has components W kl ij " trfr´B 2 f ij Bu k Bu l¯; see [9].…”

Homogeneous Levi non-degenerate hypersurfaces in $${{\mathbb {C}}}^3$$

“…4 for the definition). This corresponds to the Levi definite real form of case D.6-1 in [9]-see also Table 8. The symmetry algebra here is isomorphic to the semidirect product of sl(2, R) and the 3-dimensional Heisenberg algebra.…”

“…The corresponding complete systems of second order PDEs are encoded as complex analytic manifolds of dimension 2n − 1 equipped with contact distribution C decomposed into the direct sum of two completely integrable subdistributions E and V . Such structures are called integrable Legendrian contact (ILC) structures and they were studied in detail in [9]. The fundamental invariant obstructing flatness 2 is harmonic curvature κ H , which is a binary quartic field when n = 3, i.e.…”

“…All multiply-transitive ILC structures in dimension five (in particular having symmetry algebra of dimension ≥ 6) were classified in [9] and organized according to their Petrov type. In particular, only types III (triple root), D (two double roots), N (quadruple root), or O (flat case) arise.…”