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. 2010 MSC: Primary 53C05, Secondary 46C50. Keywords and Phrases: 2-Riemannian metric, 2-inner Product, Stationary Vector field. CM was partially supported by CNPq, FAPERJ and PRONEX/DYN-SYS. from Brazil. 2-Riemannian metricsIn this section we define 2-Riemannian metrics on a differentiable manifold. Previously we shall recall the definition of 2-inner product spaces [7].Definition 2.1. A 2-inner product in a vector space V is a map g : V × V × V → R which satisfies the following properties for all u, u ′ , v, w ∈ V and α ∈ R:(1) g(u, u/v) ≥ 0 and g(u, u/v) = 0 if and only if u and v are linearly dependent;(2) g(u, u/v) = g(v, v/u);(3) g(u, v/w) = g(v, u/w);(4) g(αu + u ′ , v/w) = αg(u, v/w) + g(u ′ , v/w).