2004/09/17 by Alfonso Gómez-Lobo, Gómez-Lobo, Alfonso García-Parrado
Mathematics · Physics and Astronomy · #53A45 #53A55 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.math-ph/0409037
openalex publication_date 2004/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper a thorough study of the normal form and the first integrability conditions arising from \em bi-conformal vector fields is presented. These new symmetry transformations were introduced in \em Class. Quantum Grav.21, 2153-2177 and some of their basic properties were addressed there. Bi-conformal vector fields are defined on a pseudo-Riemannian manifold through the differential conditions \lie Pab=ϕPab and \lieΠab=χΠab where Pab and Πab are orthogonal and complementary projectors with respect to the metric tensor \rmgab. One of the main results of our study is the discovery of a new geometric characterization of \em conformally separable spaces with conformally flat leaf metrics similar to the vanishing of the Weyl tensor for conformally flat metrics. This geometric characterization seems to carry over to any pseudo-Riemannian manifold admitting conformally flat foliations which would open the door to the systematic searching of these type of foliations in a given pseudo-Riemannian metric. Other relevant aspects such as the existence of invariant tensors under the finite groups generated by these transformations are also addressed.