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
SN - 0926-2245
VL - 31
SP - 557
EP - 567
JO - Differential Geometry and its Application
JF - Differential Geometry and its Application
IS - 5
ER -