Conformally related Riemannian metrics with non-generic holonomy

Abstract

AbstractIn this paper we show that if a compact connected n-dimensional manifold M has a conformal class containing two non-homothetic metrics g and {\tilde{g}=e^{2\varphi}g} with non-generic holonomy, then after passing to a finite covering, either {n=4} and {(M,g,\tilde{g})} is an ambikähler manifold, or {n\geq 6} is even and {(M,g,\tilde{g})} is obtained by the Calabi Ansatz from a polarized Hodge manifold of dimension {n-2}, or both g and {\tilde{g}} have reducible holonomy, M is locally diffeomorphic to a product {M_{1}\times M_{2}\times M_{3}}, the metrics g and {\tilde{g}} can be written as{g=g_{1}+g_{2}+e^{-2\varphi}g_{3}}\quad\text{and}\quad{\tilde{g}=e^{2\varphi}(% g_{1}+g_{2})+g_{3}}for some Riemannian metrics {g_{i}} on {M_{i}}, and φ is the pull-back of a non-constant function on {M_{2}}.