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arXiv · 1201.2089

2-Riemannian manifolds

C. A. Morales·M. Vilches
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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.

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C. A. Morales, M. Vilches. 2012-01-10. 2-Riemannian manifolds. https://arxiv.org/abs/1201.2089

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