2-Riemannian manifolds

Details 2-Riemannian manifolds 2-Riemannian manifolds

2012-01-10

arxiv.org/abs/1201.2089v1

SUT J. Math. 46 (2010), no. 1, 119--153

Mathematics

Differential Geometry

28 pages

Scientific paper

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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