Canonical symplectic structures on the r-th order tangent bundle of a symplectic manifold

Abstract

We describe all canonical 2-forms Ʌ(ω) on the r-th order tangent bundle TʳM = Jʳ ₀ (R, M) of a symplectic manifold (M; ω). As a corollary we deduce that all canonical symplectic structures Ʌ(ω) on TʳM over a symplectic manifold (M; ω) are of the form Ʌ(ω) = Σₖʳ =₀ αₖω(ᴷ) for all real numbers αₖ with αᵣ ≠ 0, where ω(ᴷ) is the (k)-lift (in the sense of A. Morimoto) of ω to TʳM.