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 five-dimensional integrable Legendrian contact structures whose symmetry algebra is transitive on the manifold and has at least one-dimensional isotropy algebra at any point.
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Notes
One inadvertent omission from Cartan’s list was recently discovered in [10].
In terms of \(\mathfrak {sl}_4\) fundamental weights \(\{ \lambda _i \}\), \({\mathbb {W}}\) has lowest weight \(3\lambda _1 - 4\lambda _2 + 3\lambda _3 = \alpha _1 - \alpha _2 + \alpha _3\) by the “minus lowest weight” convention [1].
This normalization is always possible working over \({{\mathbb {C}}}\), but over \({\mathbb {R}}\) we would have two possibilities: \(A = \pm {\mathsf {s}}^4\).
Implicitly, this trichotomy depends on \(B_3\) and \(B_7\) have locally constant type, i.e., the stated invariant conditions are true locally. For (multiply) transitive structures, this is always true.
The latter two correspond to \((\mathsf {r}, \mathsf {s}) \mapsto (\mathsf {r}, -\mathsf {s})\) and \((\mathsf {r}, \mathsf {s}) \mapsto (-\mathsf {s}, \mathsf {r})\).
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Acknowledgements
The Cartan and DifferentialGeometry packages in Maple (written by Jeanne Clelland and Ian Anderson respectively) provided an invaluable framework for implementing the Cartan reduction method and subsequently carrying out the analysis of the structures obtained. The work of the second and third authors was supported by ARC Discovery Grants DP130103485 and DP110100416 respectively. D.T. was also supported by Project M1884-N35 of the Austrian Science Fund (FWF).
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Appendix: Classification Tables
Appendix: Classification Tables
See Tables 9, 10, 11, 12, 13 and 14
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Doubrov, B., Medvedev, A. & The, D. Homogeneous Integrable Legendrian Contact Structures in Dimension Five. J Geom Anal 30, 3806–3858 (2020). https://doi.org/10.1007/s12220-019-00219-x
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DOI: https://doi.org/10.1007/s12220-019-00219-x