关于挠率与非度规性的Cartan几何视角

中文摘要 AI 辅助

Élie Cartan确立了，一个$D$维嵌入流形$M$的度规和内在曲率可以通过追踪另一个同维数曲面$N$在$M$上无滑动、无扭转地滚动时的响应来确定。在时空几何的背景下，这一构造支撑了广义相对论的MacDowell-Mansouri表述。我们考虑该框架的扩展，对应于带有扭转和形状演化的形状滚动；结果表明，这些扩展分别自然地描述了挠率和非度规性。此外，还讨论了遥平行引力和对称遥平行的Cartan几何表述，以及非度规性在一阶引力表述中的进一步表现。

英文摘要

Élie Cartan established that the metric and intrinsic curvature of a $D$ dimensional embedded manifold $M$ could be determined by tracing the response of another surface $N$ of the same dimensionality, as it is rolled without slipping and twisting on $M$. In the context of spacetime geometry, this construction underpins the MacDowell-Mansouri formulation of General Relativity. We consider extensions of this framework that correspond to rolling of a shape with twisting and with shape evolution; it is shown that these naturally describe torsion and non-metricity respectively. Cartan-geometric formulations of teleparallel gravity and symmetric teleparallelism are discussed in addition to further manifestations of non-metricity in first-order formulations of gravity.

发表机构

• Scuola Superiore Meridionale（南方高等学院）
• INFN– Sezione di Napoli（那不勒斯意大利国家核物理研究所）
• University of Tartu（塔尔图大学）
• Institute of Theoretical Physics and Astrophysics, University of Gdańsk（格但斯克大学理论物理与天体物理研究所）

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