  Gravitational waves (GWs) may be produced by various mechanisms in the early
universe. In particular, if parity is violated, it may lead to the production
of parity-violating GWs. In this paper, we focus on GWs on the scale of the
large-scale structure. Since GWs induce tidal deformations of the shape of
galaxies, one can extract such GW signals by observing images of galaxies in
galaxy surveys. Conventionally the detection of such signals is discussed by
considering the three-dimensional power spectra of the $E/B$-modes. Here, we
develop a complementary new technique to estimate the contribution of GWs to
the tidal force tensor field projected on the celestial sphere, which is a
directly observable quantity. We introduce two two-dimensional vector fields
constructed by taking the divergence and curl of the projected tidal field in
three dimensions. Their auto-correlation functions naturally contain
contributions of the scalar-type tidal field. However, we find that the
divergence of the curl of the projected tidal field, which is a pseudo-scalar
quantity, is free from the scalar contribution and thus enables us to extract
GW signals. We also find that we can detect parity-violating signals in the GWs
by observing the nonzero cross-correlation between the divergence of the
projected tidal field and the curl of it. It roughly corresponds to measuring
the cross-power spectrum of E and B-modes, but these are complementary to each
other in the sense that our estimator can be naturally defined locally in
position space. Finally we present expressions of the correlation functions in
the form of Fourier integrals, and discuss the properties of the kernels
specific to the GW case, which we call the overlap reduction function,
borrowing the terminology used in the pulsar timing array experiments.
