TY - JOUR

T1 - Curvature of Poisson pencils in dimension three

AU - Izosimov, Anton

N1 - Funding Information:
This work was partially supported by the Dynasty Foundation and by the Russian Foundation for Basic Research (project no. 12-01-31497 ).

PY - 2013/10

Y1 - 2013/10

N2 - A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.

AB - A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.

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U2 - 10.1016/j.difgeo.2013.05.011

DO - 10.1016/j.difgeo.2013.05.011

M3 - Article

AN - SCOPUS:84879357306

VL - 31

SP - 557

EP - 567

JO - Differential Geometry and its Applications

JF - Differential Geometry and its Applications

SN - 0926-2245

IS - 5

ER -