  In the Palatini action of general relativity the connection and the metric
are treated as independent dynamical variables. Instead of assuming a relation
between these quantities, the desired relation between them is derived through
the Euler-Lagrange equations of the Palatini action. In this manuscript we
construct an extended Palatini action, where we do not assume any a priori
relationship between the connection, the covariant metric tensor, and the
contravariant metric tensor. Instead we treat these three quantities as
independent dynamical variables. We show that this action reproduces the
standard Einstein field equations depending on a single metric tensor. We
further show that in this formulation the cosmological constant has an
additional theoretical significance. Normally the cosmological constant is
added to the Einstein field equations for the purpose of having general
relativity be consistent with cosmological observations. In the formulation
presented here, the nonvanishing cosmological constant also ensures the
self-consistency of the theory.
