  We start from a new theory (discussed earlier) in which the arena for physics
is not spacetime, but its straightforward extension-the so called Clifford
space ($C$-space), a manifold of points, lines, areas, etc..; physical
quantities are Clifford algebra valued objects, called polyvectors. This
provides a natural framework for description of supersymmetry, since spinors
are just left or right minimal ideals of Clifford algebra. The geometry of
curved $C$-space is investigated. It is shown that the curvature in $C$-space
contains higher orders of the curvature in the underlying ordinary space. A
$C$-space is parametrized not only by 1-vector coordinates $x^\mu$ but also by
the 2-vector coordinates $\sigma^{\mu \nu}$, 3-vector coordinates $\sigma^{\mu
\nu \rho}$, etc., called also {\it holographic coordinates}, since they
describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto
the coordinate planes. A remarkable relation between the "area" derivative $\p/
\p \sigma^{\mu \nu}$ and the curvature and torsion is found: if a scalar valued
quantity depends on the coordinates $\sigma^{\mu \nu}$ this indicates the
presence of torsion, and if a vector valued quantity depends so, this implies
non vanishing curvature. We argue that such a deeper understanding of the
$C$-space geometry is a prerequisite for a further development of this new
theory which in our opinion will lead us towards a natural and elegant
formulation of $M$-theory.
