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.

中文翻译：

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薛定谔连接：从数学基础到 矢野-薛定谔宇宙学


薛定谔连接是一类特殊的仿射连接，尽管它与公制不兼容，但在自动并行传输下会保留向量的长度。在本文中，我们介绍了一种新的薛定谔连接的无坐标公式。在用微分几何的语言重新塑造它们的基本属性后，我们证明了薛定谔连接可以通过扭转、非度量或两者兼而有之来实现。然后，我们计算了 Yano-Schrödinger 几何的曲率张量，并提出了第一个具有扭转的非静态 Einstein 流形的显式示例。我们将 Raychaudhuri 方程和 Sachs 方程推广到薛定谔几何。这些连接的长度保持特性使我们能够构建 Sachs 方程的拉格朗日公式。我们还得到了一个宇宙学距离方程。在这种几何分析之后，我们基于 Yano-Schrödinger 几何，使用度量和度量仿射方法构建引力理论。对于后者，我们引入了一种新的宇宙学超流体，它将提供薛定谔几何。最后，我们在这些理论中构建了简单的宇宙学模型，并将我们的结果与观测数据以及 ΛCDM 模型进行比较。