  A theory of flame propagation in curved channels is developed within the
framework of the on-shell description of premixed flames. Employing the Green
function appropriate to the given channel geometry, an implicit integral
representation for the burnt gas velocity is constructed. It is then used to
derive an explicit expression for rotational component of the gas velocity near
the flame front by successive separation of irrotational contributions. We
prove that this separation can be performed in a way consistent with boundary
conditions at the channel walls. As a result, the unknown irrotational
component can be projected out by applying a dispersion relation, thus leading
to a closed system of equations for the on-shell fresh gas velocity and the
flame front position. These equations show that in addition to the usual
nonlocality associated with potential flows, vorticity produced by a curved
flame leads to specific nonlocal spatial and temporal influence of the channel
geometry on the flame evolution. To elucidate this influence, three special
cases are considered in more detail -- steady flame stabilized by incoming flow
in a bottle-shaped channel, quasi-steady flame, and unsteady flame with small
gas expansion propagating in a channel with slowly varying width. In the latter
case, analytical solutions of the derived equations are obtained in the first
post-Sivashinsky approximation using the method of pole decomposition.
