Canonical Poisson–Nijenhuis structures on higher order tangent bundles

Abstract

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.

VB-structures and generalizations

“…There is a huge list of various concepts of lifting tensor fields and other geometric structures from M to Ì r M (e.g. [13,25,30,42,48,60,61,62,65,69,79]), starting from the complete tangent lifts [33,34,39,46,47,64,75,76,77,78,80]. We will use mainly [63] and [48], where the descriptions of lifts (prolongations) are the same although based on different concepts.…”