  A variational principle is suggested within Riemannnian geometry, in which an
auxiliary metric and the Levi Civita connection are varied independently. The
auxiliary metric plays the role of a Lagrange multiplier and introduces
non-minimal coupling of matter to the curvature scalar. The field equations are
2nd order PDEs and easier to handle than those following from the so-called
Palatini method. Moreover, in contrast to the latter method. no gradients of
the matter variables appear. In cosmological modeling, the physics resulting
from the new variational principle will differ from the modeling using the
Palatini method.
