Schrödinger联络：从数学基础到Yano-Schrödinger宇宙学
Schrödinger Connections: From Mathematical Foundations Towards Yano-Schrödinger Cosmology
AI总结：
本文建立Schrödinger联络的无坐标几何理论，推广Raychaudhuri与Sachs方程，构建度量及度量-仿射引力理论，并给出与观测和ΛCDM比较的宇宙学模型。
摘要：
Schrödinger联络是一类特殊的仿射联络，尽管与度量不相容，却能在自平行输运下保持向量长度。本文提出一种新颖的无坐标表述来刻画Schrödinger联络。在用微分几何语言重构其基本性质后，我们证明Schrödinger联络可通过挠率、非度量性或二者共同实现。随后，我们计算了Yano-Schrödinger几何的曲率张量，并给出首个具有挠率的非静态Einstein流形的显式例子。我们将Raychaudhuri方程和Sachs方程推广到Schrödinger几何；这些联络的保长性质使我们能够构建Sachs方程的Lagrange表述。我们还得到了一个宇宙学距离方程。在此几何分析之后，我们分别采用度量方法和度量-仿射方法，建立了基于Yano-Schrödinger几何的引力理论。对于后者，我们引入一种新颖的宇宙学超流体，作为Schrödinger几何的源。最后，我们在这些理论中构建了简单的宇宙学模型，并将所得结果与观测数据以及ΛCDM模型进行比较。
英文摘要：
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 $Λ$CDM model.
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