  We develop the first steps towards an analysis of geometry on the quantum
spacetime proposed in [1]. The homogeneous elements of the universal
differential algebra are naturally identified with operators living in tensor
powers of Quantum Spacetime; this allows us to compute their spectra. In
particular, we consider operators that can be interpreted as distances, areas,
3- and 4-volumes. The Minkowski distance operator between two independent
events is shown to have pure Lebesgue spectrum with infinite multiplicity. The
Euclidean distance operator is shown to have spectrum bounded below by a
constant of the order of the Planck length. The corresponding statement is
proved also for both the space-space and space-time area operators, as well as
for the Euclidean length of the vector representing the 3-volume operators.
However, the space 3-volume operator (the time component of that vector) is
shown to have spectrum equal to the whole complex plane. All these operators
are normal, while the distance operators are also selfadjoint. The Lorentz
invariant spacetime volume operator, representing the 4- volume spanned by five
independent events, is shown to be normal. Its spectrum is pure point with a
finite distance (of the order of the fourth power of the Planck length) away
from the origin. The mathematical formalism apt to these problems is developed
and its relation to a general formulation of Gauge Theories on Quantum Spaces
is outlined. As a byproduct, a Hodge Duality between the absolute dif-
ferential and the Hochschild boundary is pointed out.
