Schrödinger 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 Schrödinger connections. After recasting
their basic properties in the language of differential geometry, we show that Schrödinger
connections can be realized through torsion, non-metricity, or both. We then calculate the
curvature tensors of Yano-Schrödinger geometry and present the first explicit example of
a non-static Einstein manifold with torsion. We generalize the Raychaudhuri and Sachs
equations to the Schrödinger 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-Schrödinger geometry, using both a metric and a metric-affine
approach. For the latter, we introduce a novel cosmological hyperfluid that will source
the Schrödinger geometry. Finally, we construct simple cosmological models within these
theories and compare our results with observational data as well as the ΛCDM model.