Metric Correspondences and Low-Rank Deformations for Conformal and \(m\)-Modified Conformal Vector Fields

Abstract

We fix a smooth vector field and a positive parameter $m$, and compare the nondegenerate metrics for which the field is conformal or $m$-modified conformal with the same prescribed smooth factor. Metrics of arbitrary signature are allowed. Passing to inverse metrics gives a linear transport equation and an explicit correspondence on the exact nondegeneracy domains. At a regular point, transverse initial data yield a signature-preserving correspondence of metric germs. For a nowhere-zero field and a Killing background in dimension at least three, every metric satisfying the modified conformal equation and differing from the background by a tensor of rank at most one has factor zero; its inverse-metric difference is $rX\otimes X$ for a unique smooth function $r$. Under a complete flow, such global deformations exist precisely when the field is null for the background and admits a smooth function $τ$ with $Xτ=1$. An auxiliary field satisfying $[X,Z]=X$ gives a second correspondence under orthogonality assumptions. Finally, for a nowhere-zero complete field, a Killing metric and a zero-factor modified conformal metric have inverse-metric difference of rank at least two whenever that rank is constant. In the rank-two case, the pair determines a unique auxiliary field. Examples distinguish the roles of dimension, completeness, constant rank, the zero-factor condition, and orthogonality.