Homogeneous Integrable Legendrian Contact Structures in Dimension Five
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Homogeneous Integrable Legendrian Contact Structures in Dimension Five Boris Doubrov1
· Alexandr Medvedev2,3 · Dennis The4,5,6
Received: 12 September 2016 © Mathematica Josephina, Inc. 2019
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. Keywords Legendrian structures · Symmetry algebra · Curvature module · Multiply transitive · Complete systems of PDEs
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Boris Doubrov [email protected] Alexandr Medvedev [email protected] Dennis The [email protected]
1
Faculty of Mathematics and Mechanics, Belarusian State University, Nezavisimosti ave. 4, 220050 Minsk, Belarus
2
School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
3
Present Address: International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy
4
Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
5
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern Platz 1, 1090 Wien, Austria
6
Present Address: Department of Mathematics and Statistics, Faculty of Science and Technology, UiT The Arctic University of Norway, N-9037 Tromsø, Norway
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B. Doubrov et al.
Mathematics Subject Classification Primary: 58J70; Secondary: 35A30 · 53A40 · 53B15 · 53D10 · 22E46
1 Introduction One of the main goals of this paper is to approach this complete classification of Levi non-degenerate real hypersurfaces in C3 that form the boundary (or rather its smooth part) of a possibly unbounded homogeneous domain in C3 . It is well-known that all such bounded domains are symmetric. This result is attributed to Cartan [9], who however never published any details on this. A more recent classification of homogeneous bounded domains up to dimension 8 is present in [14]. We note that the first non-symmetric examples of bounded homogeneous domains are due to PjateckiiShapiro [22] and appear only in dimension 4. The unbounded case, however, is much less studied. One of the examples of such domains is given by the real hypersurface Im(z + w x) ¯ + |x|4 = 0, known as the Winkelmann hypersurface [27]. This hypersurface separates C3 into two open d
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