  Schr\"odinger connections are a special class of affine connections, which
despite being metric incompatible, preserve length of vectors under
autoparallel transport. In the present paper, we introduce a novel
coordinate-free formulation of Schr\"odinger connections. After recasting their
basic properties in the language of differential geometry, we show that
Schr\"odinger connections can be realized through torsion, non-metricity, or
both. We then calculate the curvature tensors of Yano-Schr\"odinger geometry
and present the first explicit example of a non-static Einstein manifold with
torsion. We generalize the Raychaudhuri and Sachs equations to the
Schr\"odinger geometry. The length-preserving property of these connections
enables us to construct a Lagrangian formulation of the Sachs equation. We also
obtain an equation for cosmological distances. After this geometric analysis,
we build gravitational theories based on Yano-Schr\"odinger geometry, using
both a metric and a metric-affine approach. For the latter, we introduce a
novel cosmological hyperfluid that will source the Schr\"odinger geometry.
Finally, we construct simple cosmological models within these theories and
compare our results with observational data as well as the $\Lambda$CDM model.
