Weyl-Cartan-Weitzenböck gravity through Lagrange multiplier

We consider an extension of the Weyl-Cartan-Weitzenb\"ock (WCW) and teleparallel gravity in which the Weitzenb\"ock 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\"ock 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\"ock condition is imposed on the Weyl-Cartan space-time, the metric is not dynamical, and the gravitational dynamics and evolution are completely determined by the torsion tensor. We show how to resolve this difficulty and generalize the WCW model by imposing the Weitzenb\"ock condition on the action of the gravitational field through a Lagrange multiplier. The gravitational field equations are obtained which explicitly depend on the Lagrange multiplier. As a particular model we consider the case of the Riemann-Cartan space-times with zero nonmetricity which mimics the teleparallel theory. 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.