Geometrical constraints on curvature from galaxy-lensing
  cross-correlations
  Accurate constraints on curvature provide a powerful probe of inflation.
However, curvature constraints based on specific assumptions of dark energy may
lead to unreliable conclusions when used to test inflation models. To avoid
this, it is important to obtain constraints that are independent on assumptions
for dark energy. In this paper, we investigate such constraints on curvature
from the geometrical probe constructed from galaxy-lensing cross-correlations.
We study comprehensively the cross-correlations of galaxy with magnification,
measured from type Ia supernovae's brightnesses ("$g\kappa^{\rm SN}$"), with
shear ("$g\kappa^{\rm g}$"), and with CMB lensing ("$g\kappa^{\rm CMB}$"). We
find for the LSST and Stage IV CMB surveys, "$g\kappa^{\rm SN}$" ,
"$g\kappa^{\rm g}$" and "$g\kappa^{\rm CMB}$" can be detected with
signal-to-noise ratio $S/N=104,\ 2291,\ 1842$ respectively. When combined with
supernovae Hubble diagram ("SN") to constrain curvature, we find galaxy-lensing
cross-correlation becomes increasingly important with more degrees of freedom
allowed in dark energy. Without any priors, we obtain error on $\Omega_K$ of
$0.723$ from "SN + $g\kappa^{\rm SN}$", $0.0417$ from "SN + $g\kappa^{\rm g}$",
and $0.04$ from "SN + $g\kappa^{\rm g}$ + $g\kappa^{\rm CMB}$" for the LSST and
Stage IV CMB surveys. The last one is more competitive than a Stage IV BAO
survey ("BAO"). When galaxy-lensing cross-correlations are added to the
combined probe of "SN + BAO + CMB", where "CMB" stands for Planck measurement
for the CMB acoustic scale, we obtain constraint on $\Omega_K$ of $0.0013$,
which is a factor of 7 improvement from "SN + BAO + CMB". We study improvements
in these results from increasing the high redshift extension of supernovae.
