JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2015.7.389 c American Institute of Mathematical Sciences Volume 7, Number 3, September 2015 pp. 389–394 A NOTE ON 2-PLECTIC HOMOGENEOUS MANIFOLDS Mohammad Shafiee Department of Mathematics Vali-e-Asr University of Rafsanjan Rafsanjan, P.O.Box 518, Iran (Communicated by Manuel de Le´ on) Abstract. In this note we study the existence of 2-plectic structures on ho- mogenous spaces. In particular we show that S 5 = SU(3) SU(2) , SU(3) S 1 , SU(3) T 2 and SO(4) S 1 admit a 2-plectic structure. Furthermore, If G is a Lie group with Lie algebra g and R is a closed Lie subgroup of G corresponding to the nilradical of g, then G R is a 2-plectic manifold. 1. Introduction. Let V be a real vector space. A 3-form ω ∈∧ 3 V ∗ is called a 2-plectic form if ω is nondegenerate in the sense that ι v ω = 0 if and only if v = 0. If ω is a 2-plectic form on V , the pair (V,ω) is called a 2-plectic vector space. A smooth manifold M is called a 2-plectic manifold if there is a closed 3-form ω on M such that (T x M,ω x ) is a 2-plectic vector space, for all x ∈ M . 2-plectic structures (and in general multisymplectic structures) in the above sense, appeared in [3] for the first time. In the same paper, The authors introduced three important classes of 2-plectic manifolds as follows: 1. Compact semisimple Lie groups with the 2-plectic structure induced by the Killing form. 2. The bundle of exterior 2-forms E on a smooth manifold M with the 2-plectic structure ω = dΘ, where Θ is the canonical 2-form on E, characterized by α ∗ (Θ) = α, for all 2-forms α on E. 3. Cosymplectic manifolds (M,θ,η) of dimension 2n + 1 with the 2-plectic structure ω = θ ∧ η, where θ is a closed 2-form and η is a closed 1-form on M such that θ n ∧ η = 0. Use of multisymplectic structures in the covariant Hamiltonian formulation of classical field theories have already been considered extensively ([6, 5, 4, 2]). In particular, 2-plectic manifolds are used to describe a classical string ([1, 9]). How- ever, geometrically, 2-plectic geometry, in contrast to symplectic case, have not been considered. A possible reason is that the Darboux theorem does not hold in this case ([10, 8]). Another reason, can be the fact that 2-plectic manifolds are not, in comparison to symplectic manifolds, do not enjoy variety. In this note, using homogeneous spaces, we try to introduce new 2-plectic manifolds. 2010 Mathematics Subject Classification. 53D05, 53C30, 11F22. Key words and phrases. 2-plectic structure, homogeneous space. The author is supported by the research Council of Vali-e-Asr University. 389