Weyl-Cartan-Weitzenb\"ock gravity through Lagrange multiplier

We consider an extension of the Weyl-Cartan-Weitzenb\"{o}ck (WCW) and teleparallel gravity, in which the Weitzenb\"{o}ck condition of the exact cancellation of curvature and torsion in a Weyl-Cartan geometry is inserted into the gravitational action via a Lagrange multiplier. In the standard metric formulation of the WCW model, the flatness of the space-time is removed by imposing the Weitzenb\"{o}ck condition in the Weyl-Cartan geometry, where the dynamical variables are the space-time metric, the Weyl vector and the torsion tensor, respectively. However, once the Weitzenb\"{o}ck condition is imposed on the Weyl-Cartan space-time, the metric is not dynamical, and the gravitational dynamics and evolution is completely determined by the torsion tensor. We show how to resolve this difficulty, and generalize the WCW model, by imposing the Weitzenb\"{o}ck condition on the action of the gravitational field through a Lagrange multiplier. The gravitational field equations are obtained from the variational principle, and they explicitly depend on the Lagrange multiplier. As a particular model we consider the case of the Riemann-Cartan space-times with zero non-metricity, which mimics the teleparallel theory of gravity. The Newtonian limit of the model is investigated, and a generalized Poisson equation is obtained, with the weak field gravitational potential explicitly depending on the Lagrange multiplier and on the Weyl vector. The cosmological implications of the theory are also studied, and three classes of exact cosmological models are considered.