Geometry of almost Cliffordian manifolds: classes of subordinated connections

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.