A direct approach to prescribing scalar curvature on bundles

Abstract

This note intends to demonstrate how to discuss scalar curvature functions' admissibility on bundles by directly applying some of the Kazdan--Warner results. Proofs of the concept include determining which functions are realizable as scalar curvature functions on the total space of several bundles over exotic manifolds. In addition, we employ traditional variational methods to provide reasonable conditions to smooth functions on the total spaces of some fiber bundles to be realized as scalar curvature functions for some Riemannian submersions metrics. We apply these last results to bundles over Calabi--Yau manifolds.