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Equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds

We solve the equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds via Cartan’s method of equivalence. The problem separates into two branches on total space, one of which ends up with the intransitive involutive structure equations. For the transitive case, we obtain an {e}-structure on both total and base spaces. | Turk J Math (2018) 42: 2452 – 2465 © TÜBİTAK doi:10.3906/mat-1708-33 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds Tuna BAYRAKDAR∗, Abdullah Aziz ERGİN Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey Received: 21.08.2017 • Accepted/Published Online: 12.07.2018 • Final Version: 27.09.2018 Abstract: We solve the equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds via Cartan’s method of equivalence. The problem separates into two branches on total space, one of which ends up with the intransitive involutive structure equations. For the transitive case, we obtain an {e} -structure on both total and base spaces. Key words: Bi-Hamiltonian structure, Poisson structure, Cartan’s method of equivalence, intransitive structure equations, Maurer–Cartan equations 1. Introduction The equivalence problem is whether two differential geometric objects on a manifold could be transformed into each other via a class of diffeomorphisms, which are obtained as the solution of a certain system of differential equations. The first known example of this problem is Poincare’s proof of the fact that two hypersurfaces of three real dimensions in C2 may fail to be biholomorphically equivalent [20]. In 1932 Cartan solved the equivalence problem of two hypersurfaces by finding local invariants [2], and then this solution was developed and generalized into a method that can be applied to the solution of various equivalence problems. For a more detailed historical account of the subject we refer to [6]. Cartan’s method of equivalence can be applied not only to geometric objects but also to various mathematical structures on a manifold [19], including differential equations. Cartan showed that it is possible to define a projective connection whose geodesics are the .