2-Riemannian manifolds

TL;DR

This paper introduces the concept of 2-Riemannian manifolds, explores their flatness conditions, defines a curvature using a unique pseudoconnection, and studies properties like stationary vector fields.

Contribution

It develops the theory of 2-Riemannian manifolds, including flatness criteria, a unique compatible pseudoconnection, and curvature properties, advancing differential geometry.

Findings

Characterization of flat 2-Riemannian manifolds

Existence of a unique torsion-free compatible pseudoconnection

Stationary vector fields in  with respect to Euclidean-induced 2-Riemannian metric are divergence-free

Abstract

A {\em 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 $\mathbb{R}^2$ with respect to the 2-Riemannian metric induced by the Euclidean product are the divergence free ones.