About curvature, conformal metrics and warped products

Details About curvature, conformal metrics and warped products About curvature, conformal metrics and warped products

2007-04-04

arxiv.org/abs/0704.0595v1

J.Phys.A40:13907-13930,2007

Mathematics

Differential Geometry

32 pages, 3 figures

Scientific paper

10.1088/1751-8113/40/46/006

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 $w^{2 \mu}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth functions and $\mu$ 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.

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