Nonmetricity on Riemann–Cartan–Weyl manifold: Its physical and mathematical meaning and application

Abstract

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.