defined on the base of the (usual) Hilbert space of quantum mechanics or its modifications. In these works different parallel transports in the corresponding bundles are introduced too.

A general feature of all of the references above-cited is that in them all geometric concepts are introduced by using in one or the other way the accepted mathematical foundation of quantum mechanics, viz.  a suitable Hilbert or projective Hilbert space and operators acting in it. The Hilbert space may be extended in a certain sense or replaced by a more general space, but this does not change the main ideas. One of the aims of this work is namely to change this mathematical background of quantum mechanics.

Separately we have to mention the approach of Prugovečki to the quantum theory, a selective summary of which can be found in [@Drechsler/Tuckey-96] (see also the references therein) and in [@Coleman-96]. It can be characterized as ‘stochastic’ and ‘bundle’. The former feature will not be discussed in the present investigation; thus we lose some advantages of the stochastic quantum theory to which we shall return elsewhere. The latter ‘part’ of the Prugovečki’s approach has some common aspects with our present work but, generally, it 