$\omega =-1$ crossing in quintessence models in Lyra's geometry

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 $-1 \leq \omega \leq 1$.