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Let M be a smooth manifold of dimension m > 0, and denote by Scan the canonical Nijenhuis tensor on T M . Let Π be a Poisson bivector on M and Π T the complete lift of Π on T M . In a previous paper, we have shown that (T M, Π T , Scan) 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 (eα), we obtain for each 0 ≤ α ≤ r the canonical Nijenhuis tensor Sα on T r M defined by the vector eα. The tensor Sα 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) , Sα) 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.

Abstract. The notion of a Euler vector field is usually defined on the tangent bundle of a finite dimensional manifold M . In this paper, we generalize this notion to the Weil bundle T A M , for any Weil algebra A and we study some properties.

Let (M, ω) be a symplectic manifold induced by an integrable G-structure P on M . In this paper, we characterize the symplectic manifolds induced by the tangent lifts of higher order r ≥ 1 of G-structure P, from M to TrM .