  We study the cosmology of quintessence models in an extended theory of
gravity in Lyra's geometry. By analyzing the possible interactions between the
quintessence scalar and the intrinsic displacement field in Lyra's geometry, we
obtain the closed form solutions of the modified Friedmann equations for four
classes of quintessence models. Though the presence of the geometrical
displacement field promises the possibility for the effective equation of state
$\omega$ of the quintessence-displacement mixture crossing the cosmological
constant boundary, the reliable quintessence scenarios in Lyra's geometry with
stable perturbation modes are still those in which $-1 \leq \omega \leq 1$.
