  In this paper, we try to understand the geometry for a nonlocal nonlinear
Schr\"{o}dinger equation (nonlocal NLS) and its discrete version introduced by
Ablowitz and Musslimani. We show that, under the gauge transformations, the
nonlocal focusing NLS and the nonlocal defocusing NLS are, respectively, gauge
equivalent to a Heisenberg-like equation and a modified Heisenberg-like
equation, and their discrete versions are, respectively, gauge equivalent to a
discrete Heisenberg-like equation and a discrete modified Heisenberg-like
equation. From the gauge equivalence, although the geometry related to the
nonlocal NLS is not very clear, we can see that the properties between the
nonlocal NLS and its discrete version and NLS and discrete NLS have big
difference. By constructing the Darboux transformation for discrete nonlocal
NLS equations including the cases of focusing and defocusing, we derive their
discrete soliton solutions, which differ from the ones obtained by using the
scattering transformation.
