2003/11/30 by Alfonso García-Parrado, José M. M. Senovilla, José M M Senovilla · 27 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Conformal map #Control and Dynamics of Mobile Robots #Curse of dimensionality #Homogeneous space #Killing vector field #Manifold (fluid mechanics) #Metric (unit) #Nonlinear Waves and Solitons #Null vector #Tensor (intrinsic definition) #Vector field #Vector space #gr-qc #math-ph #math.DG #math.MP
paper · pdf · doi:10.1088/0264-9381/21/8/017
published in Classical and Quantum Gravity 21(8), 2153-2177 (IOP Publishing) · Replaced version with some changes in the terminology and a new theorem. To appear in Classical and Quantum Gravity
arxiv created 2004/03/25 · openalex publication_date 2004/03/25 · openalex created_date 2016/06/24 · arxiv updated 2016/08/16 · openalex updated_date 2026/08/05
We introduce the concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric \G to be scaled by different conformal factors. In particular, we study their infinitesimal version, called bi-conformal vector fields. We show the differential conditions characterizing them in terms of a "square root" of the metric, or equivalently of two complementary orthogonal projectors. Keeping these fixed, the set of bi-conformal vector fields is a Lie algebra which can be finite or infinite dimensional according to the dimensionality of the projectors. We determine (i) when an infinite-dimensional case is feasible and its properties, and (ii) a normal system for the generators in the finite-dimensional case. Its integrability conditions are also analyzed, which in particular provides the maximum number of linearly independent solutions. We identify the corresponding maximal spaces, and show a necessary geometric condition for a metric tensor to be a double-twisted product. More general ``breakable'' spaces are briefly considered. Many known symmetries are included, such as conformal Killing vectors, Kerr-Schild vector fields, kinematic self-similarity, causal symmetries, and rigid motions.