  The second-order vector mode is inevitably induced from the coupling of
first-order scalar modes in cosmological perturbation theory and might hinder a
possible detection of primordial gravitational waves from inflation through
21cm lensing observations. Here, we investigate the weak lensing signal in 21cm
photons emitted by neutral hydrogen atoms in the dark ages induced by the
second-order vector mode by decomposing the deflection angle of the 21cm
lensing signal into the gradient and curl modes. The curl mode is a good tracer
of the cosmological vector and tensor modes since the scalar mode does not
induce the curl one. By comparing angular power spectra of the 21cm lensing
curl mode induced by the second-order vector mode and primordial gravitational
waves whose amplitude is parametrized by the tensor-to-scalar ratio $r$, we
find that the 21cm curl mode from the second-order vector mode dominates over
that from primordial gravitational waves on almost all scales if $r \lesssim
10^{-5}$. If we use the multipoles of the power spectrum up to $\ell_{\rm max}
= 10^{5}$ and $10^{6}$ in reconstructing the curl mode from 21cm temperature
maps, the signal-to-noise ratios of the 21cm curl mode from the second-order
vector mode achieve ${\rm S/N} \approx 0.46$ and $73$, respectively.
Observation of 21cm radiation is, in principle, a powerful tool to explore not
only the tensor mode but also the cosmological vector mode.
