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Contact Schwarzian derivatives. (English) Zbl 1093.53082

This paper is concerned with a certain notion of generalized Schwarzian derivative of contactomorphisms of contact projective structures. The latter ones were introduced by the author in [“Contact projective structures”, Indiana Univ. Math. J. 54, No. 6, 1547–1598 (2005; Zbl 1093.53083)] \((\ast)\) generalizing ideas of H. Sato and T. Ozawa published in [Contact transformations and their Schwarzian derivatives, Adv. Stud. Pure Math. 37, 337–366 (2002; Zbl 1093.53084)] and H. Sato [Schwarzian derivatives of contact diffeomorphisms, Lobachevskii J. Math. 4, 89–98 (1999; Zbl 1044.53516)].
Roughly speaking, a contact projective structure is a contact manifold \((M,H)\), where \(M\) is a manifold of dimension \(2n-1\) and \(H\subset TM\) a non integrable distribution of codimension 1, together with a \((4n-5)\)-parameter family of unparametrized curves tangent to \(H\), called contact geodesics, such that for any \(x\in M\) and any 1-dimensional subspace \(L\subset{}T_xM\) there is a unique contact path through \(x\) tangent to \(L\). Moreover, this family has to be a subset of the set of unparametrized geodesics of some linear connection \(\nabla\) on \(M\). In \((\ast)\) the author proved that such \(\nabla\) is unique if certain conditions are added. Therefore, a contact projective structure is considered as a triple \(\text{ CPS}=\left(M,H,\left[\nabla\right]\right)\), where \(\left[\nabla\right]\) is a class of projectively related connections with the properties of above.
By some of its elementary properties the classical Schwarzian derivative is identified as a cocycle of the group of diffeomorphisms of the real projective line with values in the module of quadratic differentials, vanishing on the subgroup of projective transformations. The \(n\)-dimensional Schwarzian derivative, as described for instance in [V. Ovsienko and S. Tabachnikov, Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge Tracts in Mathematics 165, Cambridge University Press (2005; Zbl 1073.53001)], generalizes this idea to that of a cocycle of the group of diffeomorphisms of an arbitrary manifold \(M\) endowed with a projective structure \(P=\left[\nabla\right]\), taking values in the module of sections of a certain tensor bundle over \(M\), vanishing on automorphisms of \(P\). The basic idea is to consider for a diffeomorphism \(\phi\) of \(M\) the difference tensor \(D={}^\phi\nabla-\nabla\) between a linear connection \(\nabla\) belonging to \(P\) and the \(\phi\)-transform \({}^\phi\nabla\) of \(\nabla\).
The aim of this paper is to analyse an analogous notion for contact projective structures. Namely, given a contact projective structure \(\text{ CPS}=\left(M,H,\left[\nabla\right]\right)\), to any automorphism \(\phi\) of the contact structure \((M,H)\) is associated a section \({ S}_{\left[\nabla\right]}(\phi)\) of a bundle \({\mathcal C}\subset\bigotimes^3(H^\ast)\), where \({\mathcal C}\) is defined by suitable symmetry and trace conditions. Again, \({ S}_{\left[\nabla\right]}(\phi)\) satisfies the cocycle condition \[ { S}_{\left[\nabla\right]}(\phi\circ\psi)= \psi^\ast\left({ S}_{\left[\nabla\right]}(\phi)\right)+ { S}_{\left[\nabla\right]}(\psi) \] for any two contactomorphisms \(\phi,\psi\) and vanishes if \(\phi\) is an automorphism of \(\text{ CPS}\).
\({ S}_{\left[\nabla\right]}(\phi)\) measures the extent to which \(\phi\) failes to be an automorphism of \((M,H)\). For the flat contact projective structure and a contactomorphism \(\phi\) of \({\mathbb R}^{2n-1}\), \({ S}_{\left[\nabla\right]}(\phi)\) is a tensor that vanishes if and only if \(\phi\) is an element of the symplectic group acting by linear fractional transformations.

MSC:

53D10 Contact manifolds (general theory)
53B10 Projective connections
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:

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