# About curvature, conformal metrics and warped products

## TL;DR

This paper investigates the curvature properties of warped product manifolds with conformal metrics, deriving formulas for Ricci and scalar curvature to analyze Einstein and constant scalar curvature conditions.

## Contribution

It provides explicit curvature formulas for warped products with conformal metrics, enabling analysis of Einstein and scalar curvature solutions in pseudo-Riemannian settings.

## Key findings

- Derived Ricci and scalar curvature formulas for warped products with conformal metrics.
- Established conditions for existence of Einstein and constant scalar curvature structures.
- Analyzed nonlinear PDEs related to curvature conditions in Riemannian cases.

## 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 $w^{2 \mu}g_B \oplus w^2 g_F$, 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.