Weyl connection

Template:Short description 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 in an attempt to unify general relativity and electromagnetism. His approach, although it did not lead to a successful theory, lead to further developments of the theory in conformal geometry, including a detailed study by Élie Cartan . They were also discussed in .

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

If $\mathrm{\nabla }$ is a Weyl connection and $h=e^{2\gamma }g$, then $\mathrm{\nabla }h=(2d\gamma +{\alpha }_g)\otimes h$ so the one-form transforms by ${\alpha }_{e^{2\gamma }g}=2d\gamma +{\alpha }_g.$ 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 $[g]$, with ${\alpha }_g=0$. This is not the most general case, however, as any such Weyl connection has the property that the one-form ${\alpha }_h$ is closed for all $h$ 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 $d{\alpha }_g$, which is independent of $g$ in the conformal class, because the difference between two ${\alpha }_g$ 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.

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

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 .

A Weyl manifold is a manifold admitting a global Weyl connection. The global analysis of Weyl manifolds is actively being studied. For example, 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.

Weyl connections have been generalized to the setting of parabolic geometries, of which conformal geometry is a special case, in .

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