Schr\"odinger Connections: From Mathematical Foundations Towards Yano-Schr\"odinger Cosmology

Abstract

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.