Geometry of almost Cliffordian manifolds: classes of subordinated connections

TL;DR

This paper explores the geometric structures of almost Clifford and almost Cliffordian manifolds, establishing the existence and uniqueness of certain connections and describing classes of distinguished connections.

Contribution

It introduces a detailed analysis of connections on Cliffordian manifolds, highlighting the richer structure compared to Clifford manifolds and explicitly characterizing distinguished connections.

Findings

Unique subordinated connection exists for almost Clifford manifolds.

Almost Cliffordian manifolds admit a richer class of distinguished connections.

Explicit description of classes of distinguished connections in Cliffordian case.

Abstract

An almost Clifford and an almost Cliffordian manifold is a $G$--structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group $GL(km, \mathbb R)$ to $GL(m, {\mathcal O})$, where $k=2^{s+t}$ and $m \in \mathbb N$. An almost Cliffordian manifold is given by a reduction of the structure group to $GL(m, \mathcal O) GL(1,\mathcal O)$. We prove that an almost Clifford manifold based on $\mathcal O$ is such that there exists a unique subordinated connection, while the case of an almost Cliffordian manifold based on $\mathcal O$ is more rich. A class of distinguished connections in this case is described explicitly.