About curvature, conformal metrics and warped products

Fernando Dobarro; Bulent Unal

doi:10.1088/1751-8113/40/46/006

paper

About curvature, conformal metrics and warped products

arXiv:0704.0595 · doi:10.1088/1751-8113/40/46/006

Abstract

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds $(B, g_{B})$ and $(F, g_{F})$ furnished with metrics of the form $c^{2} g_{B} \oplus w^{2} g_{F}$ and, in particular, of the type $w^{2 μ} g_{B} \oplus w^{2} g_{F}$ , where $c, w : B \to (0, \infty)$ are smooth functions and $μ$ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If $(B, g_{B})$ is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave-convex nonlinearities and singular partial differential equations of the Lichnerowicz-York type among others.

arXiv comment32 pages, 3 figures

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