Schrödinger Connections: From Mathematical Foundations Towards Yano-Schrödinger Cosmology

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 $\Lambda$CDM model.

Matches the accepted version in CQG

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Cited by in corpus (4)

• Length-preserving biconnection gravity and its cosmological implications
• Warm Inflation in $f(Q)$ gravity
• Geometric formulation of $k$-essence and late-time acceleration
• Yano-Schrödinger Hyperfluid: Cosmological Implications