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.