Multiply warped products with harmonic Weyl tensor and zero radial Weyl curvature

RESUMO

In this article, we study multiply warped products with a one-dimensional base, harmonic Weyl tensor and zero radial Weyl curvature. We prove that these manifolds have at most three fibers and describe their local geometry. The proof uses a conformal change which makes the warping functions quadratic polynomials, together with the compatibility equations imposed by the Ricci tensor. The resulting metrics are either obtained from locally conformally flat multiply warped products by replacing their space-form fibers with Einstein manifolds, or belong to a three-fiber family with equal fiber dimensions and positive Einstein constants. Examples show that the bound is sharp and that neither Weyl condition implies the other. We also discuss the stronger restrictions arising from soliton-type equations.