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By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show… (More)

- Pierre Bieliavsky, Michel Cahen, Simone Gutt, John Rawnsley, Lorenz Schwachhöfer
- 2006

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching… (More)

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the… (More)

- Q.-S Chi, S A Merkulov, L J Schwachhöfer
- 1996

It is proved that the Lie groups E (5) 7 and E (7) 7 represented in R 56 and the Lie group E C 7 represented in R 112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of… (More)

Given the Euclidean space R 2n+2 endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen [1] have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold (M, ω) of dimension 2n (n ≥… (More)

The real form Spin(6, H) ⊂ End(R 32) of Spin(12, C) ⊂ End(C 32) is absolutely irreducible and thus satisfies the algebraic identities (40) and (41). Therefore, it also occurs as an exotic holonomy and the associated super-manifold M g admits a SUSY-invariant polynomial. This real form has been erraneously omitted in our paper. Also, the two real… (More)

Bryant [Br] proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger [Ber]. These connections occur on moduli spaces Y of rational contact curves in a contact threefold W. Therefore, they are naturally contained in the moduli space Z of all rational curves in W. We… (More)