  We investigate the ground-state Riemannian metric and the cyclic quantum
distance of an inhomogeneous quantum Ising spin-1/2 chain in a transverse
field. This model can be diagonalized by using a general canonical
transformation to the fermionic Hamiltonian mapped from the spin system. The
ground-state Riemannian metric is derived exactly on a parameter manifold ring
$S^1$, which is introduced by performing a gauge transformation to the spin
Hamiltonian through a twist operator. The ground-state cyclic quantum distance
and the second derivative of the ground-state energy are studied in different
inhomogeneous exchange coupling parameter region. Particularly, we show that
the quantum ferromagnetic phase in the uniform Ising chain can be characterized
by an invariant cyclic quantum distance with a constant ground-state Riemannian
metric, and this metric will rapidly decay to zero in the paramagnetic phase.
