Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba

Annales Polonici Mathematici, Tome 111 (2014), p. 21-37 / Harvested from The Polish Digital Mathematics Library

Résumé

Let M be a smooth manifold of dimension m>0, and denote by $S_{can}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^{T}$ the complete lift of Π on TM. In a previous paper, we have shown that $(TM, Π^{T}, S_{can})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^{r}M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^{A}M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002), 243-257], where A is a Weil algebra. In the particular case where $A= J^{r}₀(ℝ, ℝ) ≃ ℝ^{r+1}$ with the canonical basis $(e_{α})$, we obtain for each 0 ≤ α ≤ r the canonical Nijenhuis tensor $S_{α}$ on $T^{r}M$ defined by the vector $e_{α}$. The tensor $S_{α}$ is called the canonical Nijenhuis tensor on $T^{r}M$ of degree α. In this paper, we show that if (M,Π) is a Poisson manifold, then for each α with 1 ≤ α ≤ r, $(T^{r}M, Π^{(c)}, S_{α})$ is a Poisson-Nijenhuis manifold. In particular, we describe other prolongations of Poisson manifolds from M to $T^{r}M$ and we give some of their properties.

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Publié le : 2014-01-01

Zbl 1305.53038

EUDML-ID : urn:eudml:doc:280896