Schrödinger connections: from mathematical foundations towards Yano–Schrödinger cosmology

Abstract

Abstract 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.