  We study the variational principle over an Hilbert-Einstein like action for
an extended geometry taking into account torsion and non-metricity. By
extending the semi-Riemannian geometry, we obtain an effective energy-momentum
tensor which can be interpreted as physical sources. As an application we
develop a new manner to obtain the gravitational wave equations on a
Weyl-integrable manifold taking into account the non-metricity and non-trivial
boundary conditions on the minimization of the action, which can be identified
as possible sources for the cosmological constant and provides two different
equations for gravitational waves. We examine gravitational waves in a
pre-inflationary cosmological model.
