An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three

Abstract

We study the complete, compact, locally affine manifolds equipped with a k-symplectic structure, which are the quotients of Rn(k+1) by a subgroup Γ of the affine group A(n(k+1)) of Rn(k+1) acting freely and properly discontinuously on Rn(k+1) and leaving invariant the k-symplectic structure, then we construct and give some examples and properties of compact, complete, locally affine two-symplectic manifolds of dimension three.

Keywords

• <i>k</i>-symplectic structure
• locally affine manifolds
• foliations
• Lagrangian submanifolds