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A new natural structure on the tangent spaces of a co-tangent bundle is introduced and some of its properties are investigated. This structure is based on a symmetric bilinear form and leads to a geometry that is, in many respects, analogous to the symplectic geometry. The new structure can thus justifiably be called co-symplectic geometry. The null structure of co-symplectic vector spaces is investigated in detail. It is found that the manifold of all maximally isotropic subspaces of a co-symplectic vector space is a homogeneous compact manifold of dimension 1/2n(n-1) consisting of two diffeomorphic components and having fundamental groupZ2?Z2. A representation of the fundamental group of this manifold is explicitly constructed in terms of quadrupoles of co-Lagrangian subspaces.