In differential geometry, a Weyl connection (also called a Weyl structure) is a generalization of the Levi-Civita connection that makes sense on a conformal manifold. They were introduced by Hermann Weyl (Weyl 1918) in an attempt to unify general relativity and electromagnetism. His approach, although it did not lead to a successful theory,^[1] lead to further developments of the theory in conformal geometry, including a detailed study by Élie Cartan (Cartan 1943). They were also discussed in Eisenhart (1927).

Specifically, let ${\displaystyle M}$ be a smooth manifold, and ${\displaystyle [g]}$ a conformal class of (non-degenerate) metric tensors on ${\displaystyle M}$, where ${\displaystyle h,g\in [g]}$ iff ${\displaystyle h=e^{2\gamma }g}$ for some smooth function ${\displaystyle \gamma }$ (see Weyl transformation). A Weyl connection is a torsion free affine connection on ${\displaystyle M}$ such that, for any ${\displaystyle g\in [g]}$,

$${\displaystyle \nabla g=\alpha _{g}\otimes g}$$

where ${\displaystyle \alpha _{g}}$ is a one-form depending on ${\displaystyle g}$.

If ${\displaystyle \nabla }$ is a Weyl connection and ${\displaystyle h=e^{2\gamma }g}$, then ${\textstyle \nabla h=(2\,d\gamma +\alpha _{g})\otimes h}$ so the one-form transforms by ${\textstyle \alpha _{e^{2\gamma }g}=2\,d\gamma +\alpha _{g}.}$ Thus the notion of a Weyl connection is conformally invariant, and the change in one-form is mediated by a de Rham cocycle.

An example of a Weyl connection is the Levi-Civita connection for any metric in the conformal class ${\displaystyle [g]}$, with ${\displaystyle \alpha _{g}=0}$. This is not the most general case, however, as any such Weyl connection has the property that the one-form ${\displaystyle \alpha _{h}}$ is closed for all ${\displaystyle h}$ belonging to the conformal class. In general, the Ricci curvature of a Weyl connection is not symmetric. Its skew part is the dimension times the two-form ${\displaystyle d\alpha _{g}}$, which is independent of ${\displaystyle g}$ in the conformal class, because the difference between two ${\displaystyle \alpha _{g}}$ is a de Rham cocycle. Thus, by the Poincaré lemma, the Ricci curvature is symmetric if and only if the Weyl connection is locally the Levi-Civita connection of some element of the conformal class.^[2]

Weyl's original hope was that the form ${\displaystyle \alpha _{g}}$ could represent the vector potential of electromagnetism (a gauge dependent quantity), and ${\displaystyle d\alpha _{g}}$ the field strength (a gauge invariant quantity). This synthesis is unsuccessful in part because the gauge group is wrong: electromagnetism is associated with a ${\displaystyle U(1)}$ gauge field, not an ${\displaystyle \mathbb {R} }$ gauge field.

Hall (1993) harvtxt error: no target: CITEREFHall1993 (help) showed that an affine connection is a Weyl connection if and only if its holonomy group is a subgroup of the conformal group. The possible holonomy algebras in Lorentzian signature were analyzed in Dikarev (2021).

A Weyl manifold is a manifold admitting a global Weyl connection. The global analysis of Weyl manifolds is actively being studied. For example, Mason & LeBrun (2008) harvtxt error: no target: CITEREFMasonLeBrun2008 (help) considered complete Weyl manifolds such that the Einstein vacuum equations hold, an Einstein–Weyl geometry, obtaining a complete characterization in three dimensions.

Weyl connections also have current applications in string theory and holography.^[3][4]

Weyl connections have been generalized to the setting of parabolic geometries, of which conformal geometry is a special case, in Čap & Slovák (2003).

Citations

References

• Bergmann, Peter (1942), Introduction to the theory of relativity, Prentice-Hall.
• Čap, Andreas; Slovák, Jan (2003), "Weyl structures for parabolic geometries", Mathematica Scandinavica, 93 (1): 53–90, arXiv:math/0001166, doi:10.7146/math.scand.a-14413, JSTOR 24492421.
• Cartan, Élie (1943), "Sur une classe d'espaces de Weyl", Annales scientifiques de l'École Normale Supérieure, 60 (3): 1–16, doi:10.24033/asens.901.
• Ciambelli, Luca; Leigh, Robert (2020), "Weyl connections and their role in holography", Physical Review D, 101 (8): 086020, arXiv:1905.04339, doi:10.1103/PhysRevD.101.086020, S2CID 152282710
• Dikarev, A (2021), "On holonomy of Weyl connections in Lorentzian signature", Differential Geometry and Its Applications, 76 (101759), arXiv:2005.08166, doi:10.1016/j.difgeo.2021.101759, S2CID 218673884.
• Eisenhart, Luther (1927), Non-Riemannian geometry, AMS.
• Folland, Gerald (1970), "Weyl manifolds", Journal of Differential Geometry, 4 (2): 145–153, doi:10.4310/jdg/1214429379.
• Hall, G. (1992), "Weyl manifolds and connections", Journal of Mathematical Physics, 33 (7): 2633, doi:10.1063/1.529582.
• Higa, Tatsuo (1993), "Weyl manifolds and Einstein–Weyl manifolds", Commentarii Mathematici Universitatis Sancti Pauli, 42 (2): 143–160.
• Jia, W; Karydas, M (2021), "Obstruction tensors in Weyl geometry and holographic Weyl anomaly", Physical Review D, 104 (126031): 126031, arXiv:2109.14014, doi:10.1103/PhysRevD.104.126031, S2CID 238215186
• LeBrun, Claude; Mason, Lionel J. (2009), "The Einstein–Weyl equations, scattering maps, and holomorphic disks", Mathematical Research Letters, 16 (2): 291–301, arXiv:0806.3761, doi:10.4310/MRL.2009.v16.n2.a7.
• Weyl, Hermann (1918), "Reine Infinitesimalgeometrie", Mathematische Zeitschrift, 2 (3–4): 384–411, doi:10.1007/BF01199420, S2CID 186232500.

Further reading

• Matsuzoe, Hiroshi (2001), "Geometry of semi-Weyl manifolds and Weyl manifolds", Kyushu Journal of Mathematics, 55: 107–117, doi:10.2206/kyushujm.55.107.
• Pedersen, H.; Tod, K. P. (1993), "Three dimensional Einstein–Weyl geometry", Advances in Mathematics, 97 (1): 74–109, doi:10.1006/aima.1993.1002.
• Hirică, Iulia; Nicolescu, Liviu (2004), "On Weyl structures", Rendiconti del Circolo Matematico di Palermo, 53 (3): 390–400, doi:10.1007/BF02875731, S2CID 123385518.
• Jiménez, Jose; Koivisto, Tomi (2014), "Extended Gauss–Bonnet gravities in Weyl geometry", Classical and Quantum Gravity, 31 (13): 135002, arXiv:1402.1846, doi:10.1088/0264-9381/31/13/135002, S2CID 118424219.
• Čap, Andreas; Mettler, Thomas (2023), "Geometric theory of Weyl structures", Communications in Contemporary Mathematics, 25 (7): 2250026, arXiv:1908.10325, doi:10.1142/S0219199722500262, S2CID 201646408.
• Mettler, Thomas; Paternain, Gabriel (2020), "Convex projective surfaces with compatible Weyl connection are hyperbolic", Analysis & PDE, 13 (4): 1073–1097, arXiv:1804.04616, doi:10.2140/apde.2020.13.1073, S2CID 119657577.
• Alexandrov, B; Ivanov, S (2003), "Weyl structures with positive Ricci tensor", Differential Geometry and Its Applications, 18 (3): 343–350, arXiv:math/9902033, doi:10.1016/S0926-2245(03)00010-X, S2CID 119624508.
• Florin Belgun; Andrei Moroianu (2011), "Weyl-parallel forms, conformal products, and Einstein–Weyl manifolds", Asian Journal of Mathematics, 15 (4): 499–520, arXiv:0901.3647, doi:10.4310/AJM.2011.v15.n4.a1, S2CID 55210918.

See also

• Einstein–Weyl geometry

External links

• Weyl connection, Encyclopedia of Mathematics