Higher Affine Connections

Abstract

For a smooth manifold M, it was shown in Broojerdian et al. (Iran J Math Sci Inform 6(1):79–96, 2011) that every affine connection on the tangent bundle TM naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of Broojerdian et al. (Iran J Math Sci Inform 6(1):79–96, 2011) and construct covariant derivatives along MVFs which are not induced by affine connections on TM. We call this more general class of covariant derivatives higher affine connections. In addition, we also propose a framework which gives rise to non-induced higher connections; this framework is obtained by equipping the full exterior bundle \({\wedge^\bullet TM}\) with an associative bilinear form \({\eta}\). Since the latter can be shown to be equivalent to a set of differential forms of various degrees, this framework also provides a link between higher connections and multisymplectic geometry.