2024/09/25 by Maciej Dunajski, Dunajski, Maciej, A. Rod Gover +1
Chemistry · Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2409.17347
openalex publication_date 2024/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non--linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to Λ3(\cT), where \cT is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms. Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing-Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterisation of algebraically special Riemannian metrics of type D in dimensions higher than four.