2019/03/31 by Bert Janssen, Alejandro Jiménez-Cano, Alejandro Jimenez-Cano +2
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Affine connection #Affine transformation #Black Holes and Theoretical Physics #Connection (principal bundle) #Cosmology and Gravitation Theories #Curvature #Fundamental theorem of Riemannian geometry #Geometry #Invariant (physics) #Levi-Civita connection #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Metric connection #Pure mathematics #Ricci curvature #Rotation formalisms in three dimensions #gr-qc #hep-th
paper · pdf · doi:10.1016/j.physletb.2019.06.002
published as Phys. Lett. B795 (2019) 42-48 · 13 pages, 1 figure. v2: References added. v3: 14 pages. Minor changes, references added. Version to be published in Phys.Lett.B
openalex publication_date 2019/06/04 · arxiv created 2019/06/05 · arxiv updated 2019/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all connections constructed by acting with a projective transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss–Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss–Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.