Bi-conformal vector fields and their applications

TL;DR

This paper introduces bi-conformal vector fields, a generalization of conformal transformations, characterizing their properties, Lie algebra structure, and applications to various known symmetries in differential geometry.

Contribution

It defines bi-conformal vector fields, analyzes their differential conditions, Lie algebra structure, and maximal spaces, extending known symmetry concepts in geometric analysis.

Findings

Characterization of bi-conformal vector fields via a 'square root' of the metric

Determination of conditions for finite and infinite-dimensional Lie algebras

Identification of maximal spaces and geometric conditions for double-twisted products

Abstract

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.