  In the present paper we introduce a geometric generalization of standard
general relativity, based on a geometry initially introduced by Friedmann and
Schouten in 1924, through the notion of a semi-symmetric connection. The
semi-symmetric connection is a particular connection that extends the
Levi-Civita one, by allowing for the presence of torsion. While the
mathematical landscape of the semi-symmetric metric connections is
well-explored, their physical implications remain to be investigated. After
presenting in detail the differential geometric aspects of the geometries with
semi-symmetric metric connection, we formulate the Einstein field equations,
which contain additional terms induced by the presence of the specific form of
torsion we are studying. We consider the cosmological applications of the
theory by deriving the generalized Friedmann equations, which also include some
supplementary terms as compared to their general relativistic counterparts and
can be interpreted as a geometric type dark energy. To evaluate the proposed
theory, we consider three cosmological models - the first with constant
effective density and pressure, the second with the dark energy satisfying a
linear equation of state, and a third one one with a polytropic equation of
state. We compare the predictions of the semi-symmetric metric gravitational
theory with the observational data for the Hubble function, and with the
predictions of the standard $\Lambda$CDM model. Our findings indicate that the
semi-symmetric metric cosmological models give a good description of the
observational data, and for certain values of the model parameters, they can
reproduce almost exactly the predictions of the $\Lambda$CDM paradigm.
Consequently, Friedmann's initially proposed geometry emerges as a credible
alternative to standard general relativity, in which dark energy has a purely
geometric origin.
