Riemannian

Reality is composed of many dimensions, all necessary for manifest forms and our experience of them. It also reveals their hierarchic functioning, which in turn points to the absolute as the apex or center of that hierarchy. It reveals the objective relation of the absolute to manifest reality as the center of a hierarchy of manifestation, and shows us that by descending into manifestation the absolute descends into an increasing number of dimensions. It also shows that this hierarchy of manifestation is not temporal or spatial, but ontological. The integrated dimension reveals the manifest world as a Riemannian multidimensional manifold, and shows the hierarchical relationships of the functioning of its various dimensions. (Inner Journey Home, pg 433)

Riemannian Manifold

Viewing our consciousness in terms of its dimensions, we see that it is not a three-or four-dimensional field, but a multidimensional manifold, a manifold in the sense that it is a dynamic structure of dynamic structures. This manifold is characterized by a nonlinear -- Riemannian -- geometry, in that all the dimensions open up to all the others in nonlinear ways. Even in ordinary human experience, we can be aware of many different levels of experience in a given moment... All our inner experiences are manifestations of forms within the manifold of our consciousness, where these forms are nothing but the very fabric of this consciousness morphing itself at particular regions. This view is not only a more faithful rendering of our inner experience, but clearly reveals the purpose of inner work. This purpose is realizing the awareness of the field of consciousness itself, which makes it possible for us to inhabit our inner field completely in all of its richness and freedom. (Inner Journey Home, pg 51)