Higher Affine Connections

TL;DR

This paper introduces higher affine connections, a generalization of affine connections on manifolds, enabling covariant derivatives along multivector fields not derived from tangent bundle connections, and links them to multisymplectic geometry.

Contribution

It generalizes covariant derivatives to include non-induced higher connections via an exterior bundle framework with an associative bilinear form.

Findings

Constructed covariant derivatives along MVFs not induced by tangent bundle connections.

Established a framework linking higher connections with multisymplectic geometry.

Extended the concept of affine connections to a broader class of geometric structures.

Abstract

For a smooth manifold $M$, it was shown in \cite{BPH} 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 \cite{BPH} and construct covariant derivatives along MVFs which are not induced by affine connections on $TM$. We call this more general class of covariant derivatives \textit{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.