  One of the important extensions of Riemann geometry is Weyl geometry, which
is essentially based on the ideas of conformal invariance and nonmetricity. A
similar non-Riemannian geometry was proposed by Erwin Schr\"{o}dinger in the
late 1940s, in a geometry which is simpler, and (probably) more elegant than
the Weyl geometry. Even it contains nonmetricity, the Schr\"{o}dinger
connection preserves the length of vectors under parallel transport, and thus
seems to be more physical than the Weyl connection. Interestingly enough,
Schr\"{o}dinger's approach did not attract much interest in the field of
gravitational physics. It is the goal of the present paper to reconsider the
Schr\"{o}dinger geometry as a potential candidate for a gravitational theory
extending standard general relativity. We consider a gravitational action
constructed from a length preserving non-metricity, in the absence of torsion,
and investigate its variation in both Palatini and metric formalisms. While the
Palatini variation leads to standard general relativity, the metric version of
the theory adds some non-metricity dependent extra terms in the gravitational
Einstein equations, which can be interpreted as representing a geometric type
dark energy. After obtaining the generalized Friedmann equations, we analyze in
detail the cosmological implications of the theory, by considering two distinct
models, corresponding to a dark energy satisfying a linear equation of state,
and to conserved matter energy, respectively. In both cases we compare the
predictions of the Weyl-Schr\"{o}dinger cosmology with a set of observational
data for the Hubble function, and with the results of the $\Lambda$CDM standard
paradigm.
