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⊗ 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.
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BibTeX
@misc{li2026metric,
title = {{Metric Correspondences and Low-Rank Deformations for Conformal and \(m\)-Modified Conformal Vector Fields}},
author = {Zhigang Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00844},
eprint = {2611.00844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00844},
}
RIS
TY - UNPB
TI - Metric Correspondences and Low-Rank Deformations for Conformal and m-Modified Conformal Vector Fields
AU - Zhigang Li
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00844
UR - https://arxiv.org/abs/2611.00844
AB - 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⊗ 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.
ER -