2019/06/30 by Adria Delhom, Iarley P. Lobo, Gonzalo J. Olmo +1
Materials Science · Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Affine connection #Affine transformation #Class (philosophy) #Conformal map #Connection (principal bundle) #Equivalence (formal languages) #Invariant (physics) #Nonlinear Waves and Solitons #Quasicrystal Structures and Properties #Weyl tensor #Weyl transformation #gr-qc #hep-th
paper · pdf · doi:10.1140/epjc/s10052-019-7394-z
published as Eur. Phys. J. C 79:878 (2019) · 9 pages, updated to match published version, some discussions extended
openalex publication_date 2019/10/01 · arxiv created 2019/10/30 · arxiv updated 2019/11/01 · openalex created_date 2019/11/01 · openalex updated_date 2026/08/05
Abstract A Weyl structure is usually defined by an equivalence class of pairs (g, \varvecω ) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> related by Weyl transformations, which preserve the relation ∇ g=\varvecω ⊗ g <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>∇</mml:mi><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mo>⊗</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:math> , where g <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi></mml:math> and \varvecω <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ω</mml:mi></mml:mrow></mml:math> denote the metric tensor and a 1-form field. An equivalent way of defining such a structure is as an equivalence class of conformally related metrics with a unique affine connection Γ _(\varvecω ) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Γ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:msub></mml:math> , which is invariant under Weyl transformations. In a standard Weyl structure, this unique connection is assumed to be torsion-free and have vectorial non-metricity. This second view allows us to present two different generalizations of standard Weyl structures. The first one relies on conformal symmetry while allowing for a general non-metricity tensor, and the other comes from extending the symmetry to arbitrary (disformal) transformations of the metric.