  We consider a partial data inverse problem for a time-dependent
convection-diffusion equation on an admissible manifold. We prove that the
time-dependent convection term and time-dependent density can be recovered
uniquely modulo a known gauge invariance. There have been several works on
inverse problems related to the steady state convection-diffusion operator in
Euclidean as well as in Riemannian geometry settings; however, inverse problems
related to time-dependent convection-diffusion equation on a manifold are not
studied in the prior works, which is the main aim of this paper. In fact, to
the best of our knowledge, the problem studied here is the first work related
to a partial data inverse problem for recovering both first and zeroth-order
time-dependent perturbations of evolution equations in the Riemannian geometry
setting.
