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.

讨论了外尔流形上非度量性的物理意义和数学意义以及对偶仿射连接在其上的应用。从仿射连接的角度来看，Weyl流形中的Weyl 1-form被认为是非度量的，并且在平行传输下引起尺度变化且具有共形不变性。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场到场的额外耦合，从黎曼流形突出。具有共形不变性的尺度变化对应于每个流形的体积畸变的场大小的类似变化。Weyl 1-form 是黎曼流形中体积元（尺度）的变化率。所以，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失真的膨胀或收缩率。黎曼流形上作为非度量的体积畸变可以通过在黎曼-嘉当-外尔流形上设置对偶仿射连接来消除，该连接用于统计流形。并基于此概念讨论了非度量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的非度量意义。