On 2-Riemannian manifolds

Abstract

A $2$-Riemannian manifold is a differentiable manifold exhibiting a $2$-inner product on each tangent space. We first study lower dimensional $2$-Riemannian manifolds by giving necessary and sufficient conditions for flatness. Afterward we associate to each $2$-Riemannian manifold a unique torsion free compatible pseudoconnection. Using it we define a curvature for $2$-Riemannian manifolds and study its properties. We also prove that $2$-Riemannian pseudoconnections do not have Koszul derivatives. Moreover, we define stationary vector field with respect to a $2$-Riemannian metric and prove that the stationary vector fields in ${ℝ}^2$ with respect to the $2$-Riemannian metric induced by the Euclidean product are the divergence free ones.