  The fidelity per site between two ground states of a quantum lattice system
corresponding to different values of the control parameter defines a surface
embedded in a Euclidean space. The Gaussian curvature naturally quantifies
quantum fluctuations that destroy orders at transition points. It turns out
that quantum fluctuations wildly distort the fidelity surface near the
transition points, at which the Gaussian curvature is singular in the
thermodynamic limit. As a concrete example, the one-dimensional quantum Ising
model in a transverse field is analyzed. We also perform a finite size scaling
analysis for the transverse Ising model of finite sizes. The scaling behavior
for the Gaussian curvature is numerically checked and the correlation length
critical exponent is extracted, which is consistent with the conformal
invariance at the critical point.
