Nowhere conformally homogeneous manifolds and limiting Carleman weights

TL;DR

This paper demonstrates that in dimensions three and higher, generic Riemannian manifolds lack nontrivial local conformal symmetries and conformal Killing fields, implying the nonexistence of limiting Carleman weights near any point, which impacts inverse problems.

Contribution

It establishes that generic high-dimensional manifolds do not admit nontrivial local conformal diffeomorphisms or conformal Killing fields, extending Sunada's results to conformal geometry and inverse problems.

Findings

Generic manifolds of dimension ≥ 3 lack local conformal diffeomorphisms.

Such manifolds do not admit nontrivial conformal Killing vector fields.

They do not admit limiting Carleman weights near any point.

Abstract

In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analog of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symmetries. Consequently, generic manifolds of dimension $\geq 3$ do not admit nontrivial conformal Killing vector fields near any point. As an application to the inverse problem of Calder\'on on manifolds, this implies that generic manifolds of dimension $\geq 3$ do not admit limiting Carleman weights near any point.