We study the observational signature of vector metric perturbations through
the effect of weak gravitational lensing. In the presence of vector
perturbations, the non-vanishing signals for B-mode cosmic shear and curl-mode
deflection angle, which have never appeared in the case of scalar metric
perturbations, naturally arise. Solving the geodesic and geodesic deviation
equations, we drive the full-sky formulas for angular power spectra of weak
lensing signals, and give the explicit expressions for E-/B-mode cosmic shear
and gradient-/curl-mode deflection angle. As a possible source for seeding
vector perturbations, we then consider a cosmic string network, and discuss its
detectability from upcoming weak lensing and CMB measurements. Based on the
formulas and a simple model for cosmic string network, we calculate the angular
power spectra and expected signal-to-noise ratios for the B-mode cosmic shear
and curl-mode deflection angle. We find that the weak lensing signals are
enhanced for a smaller intercommuting probability of the string network, $P$,
and they are potentially detectable from the upcoming cosmic shear and CMB
lensing observations. For $P\sim 10^{-1}$, the minimum detectable tension of
the cosmic string will be down to $G\mu\sim 5\times 10^{-8}$. With a
theoretically inferred smallest value $P\sim 10^{-3}$, we could even detect the
string with $G\mu\sim 5\times 10^{-10}$.