  We explore two approaches to characterise the quantum geometry of the ground
state of correlated fermions in terms of the distance matrix in the spectral
parameter space. (a) An intrinsic geometry approach, in which we study the
intrinsic curvature defined in terms of the distance matrix. (b) An extrinsic
geometry approach, in which we investigate how the distance matrix can be
approximately embedded in finite dimensional Euclidean spaces. We implement
these approaches for the ground state of a system of one-dimensional fermions
on a 18-site lattice with nearest neighbour repulsion. The intrinsic curvature
sharply changes around the Fermi points in the metallic regime but is more or
less uniform in the insulating regime. In the metallic regime, the embedded
points clump into two well seperated sets, one corresponding to modes in the
Fermi sea and the other to the modes outside it. In the insulating regime, the
two sets tend to merge.
