symplectice

Symplectice is a term used in mathematical discourse to denote a class of geometric structures that generalize symplectic geometry. A symplectice structure on a smooth manifold M of dimension 2n consists of a closed 2-form ω of constant rank 2k, with 1 ≤ k ≤ n. If k = n and ω is nondegenerate, (M, ω) is a standard symplectic manifold. If k < n, ω is degenerate, and its kernel defines a smooth distribution whose integral leaves carry a compatible symplectic form; in this sense a symplectice manifold is foliated by leaves each carrying a symplectic structure.

Historically, the term "symplectice" is not universally standardized and appears in exploratory literature as a way to discuss symplectic-like structures that relax nondegeneracy. Some authors emphasize a connection with Poisson geometry, where ω may be interpreted as a transverse structure to a Poisson tensor, and with foliation theory, where foliations have symplectic leaves.

Key properties include closure (dω = 0), constant rank, and the local behavior governed by the Frobenius integrability condition for ker ω. The set of leaves gives a foliation of M by symplectic manifolds of dimension 2k, and, under suitable transversality conditions, one can define transverse geometric structures.

Examples range from the standard symplectic manifolds (which are symplectice with k = n) to four-dimensional near-symplectic forms where ω degenerates at isolated points. Symplectice geometry connects to Poisson manifolds, where ω corresponds to a nondegenerate bivector on symplectic leaves.

See also: Symplectic geometry; near-symplectic forms; Poisson geometry; foliation theory.