k-Symplectic Affine Manifolds

Abstract

Let M be a smooth manifold of dimension n. We say that M is an affine manifold if there is an atlas (U _i, φ _i) of M such that the changes of coordinates are restrictions of affine transformations of ∝ⁿ. An affine structure on M is equivalent to a given connection

$$\nabla \Gamma (TM) \times \Gamma (TM) \to \Gamma (TM)$$

such that both the curvature

$$k(X,Y) = {\nabla _{\left[ {X,Y} \right]}} - ({\nabla _X}{\nabla _Y} - {\nabla _Y}{\nabla _X})$$

and torsion

$$T(X,Y) = {\nabla _X}Y - {\nabla _Y}X - [X,Y]$$

vanish identically.