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Abstract

Let $M$ be a smooth manifold of dimension $m>0$, and denote by $S_{\rm can}$ the canonical Nijenhuis tensor on $TM$. Let $\varPi $ be a Poisson bivector on $M$ and $\varPi ^{T}$ the complete lift of $\varPi $ on $TM$. In a previous paper, we have shown that $(TM, \varPi ^{T}, S_{\rm can})$ is a Poisson–Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from $M$ to $T^rM$ 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 $A= J^{r}_{0}(\mathbb {R}, \mathbb {R})\simeq \mathbb {R}^{r+1}$ with the canonical basis $(e_{\alpha })$, we obtain for each $0\leq \alpha \leq r$ the canonical Nijenhuis tensor $S_{\alpha }$ on $T^{r}M$ defined by the vector $e_{\alpha }$. The tensor $S_{\alpha }$ is called the canonical Nijenhuis tensor on $T^{r}M$ of degree $\alpha $. In this paper, we show that if $(M, \varPi )$ is a Poisson manifold, then for each $\alpha $ with $1\leq \alpha \leq r$, $(T^{r}M, \varPi ^{(c)}, S_{\alpha })$ 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.