Nonmetricity on Riemann-Cartan-Weyl manifold: Its physical and mathematical meaning and application
Nonmetricity on Riemann-Cartan-Weyl manifold: Its physical and mathematical meaning and application
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Riemann-Cartan-Weyl流形上的非度量性：其物理和数学意义及应用
DOI：
10.1142/s0219887822501535
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发表时间：
2022
影响因子：
1.8
通讯作者：
Nagahama Hiroyuki
中科院分区：
文献类型：
--
作者：
Hirano Mitsuhiro;Nagahama Hiroyuki
讨论了Weyl流形非度量性的物理意义和数学意义，以及对偶仿射连接的应用。从仿射连接的角度出发，将Weyl流形中的Weyl - 1形式识别为非对称形式，并在平行移动下引起保形不变的尺度变化。在表示时空和物质空间的流形上，Weyl - 1形式等价于表示为黎曼流形的额外耦合场与场，从黎曼流形中突出。共形不变性的尺度变化对应于每个流形体积畸变的场的大小的类似变化。Weyl - 1形式表示黎曼流形中体积元（尺度）的变化率。因此，Weyl流形中的非度量性被定义为由于额外耦合场引起的类似体积畸变的膨胀率或收缩率。通过在Riemann-Cartan-Weyl流形上设置对偶仿射连接，可以消除黎曼流形上作为非度量性的体积畸变，该方法应用于统计流形。在此基础上，讨论了非度量性在对称远平行流形框架下修正引力理论中的作用，以及统计流形上非度量性的意义。
The physical meaning and mathematical meaning of nonmetricity on Weyl manifold and the application of dual affine connection for it are discussed. In viewpoint from affine connection, Weyl 1-form in Weyl manifold is recognized as nonmetricity and causes scale change with conformal invariance under parallel transport. On manifolds expressing spacetime and material space, Weyl 1-form is equivalent to the extra coupling field to field expressed as Riemannian manifold, protruding from Riemannian manifold. The scale change with conformal invariance corresponds to the similar change in the magnitude of the field with the volumetric distortion of each manifold. Weyl 1-form is the change rate of volume element (scale) in Riemannian manifold. Therefore, nonmetricity in Weyl manifold is defined as expansion or contraction rate as similar volumetric distortion due to the extra coupled field. The volumetric distortion as nonmetricity on Riemannian manifold can be canceled out by setting dual affine connection on Riemann–Cartan–Weyl manifold, which is used in statistical manifold. Moreover, the role of nonmetricity in modified gravity theory in the framework of symmetric teleparallel manifold and the meaning of nonmetricity on statistical manifold are discussed based on this concept.