Einstein Warped Products with Einstein Base and Fiber
Márcio Lemes de Sousa, Tibério Bittencourt de Oliveira Martins, Carlos Rodrigues da Silva
TL;DR
This work classifies Einstein warped product manifolds with both base and fiber Einstein. Using O'Neill formulas, it derives necessary and sufficient conditions for $M = B \times_f F$ to be Einstein, establishing $Hess_{g_B} f = \frac{f}{d}(\lambda_B - \lambda) g_B$, $\lambda = \left(1 + \frac{d}{n-1}\right) \lambda_B$, $\|\nabla f\|^2 = -\rho f^2 + c$ with $\rho = \frac{\lambda_B}{n-1}$, and $\lambda_F = (d-1) c$; it also provides corollaries characterizing Ricci-flatness and base compactness. In the hyperbolic-base case $M = (\mathbb{H}^n,g_{-1}) \times_f F^d$, the authors obtain an explicit warping function $f$ of the form $f = \frac{1}{x_n} \left(\sum_{j=1}^{n-1} \frac{a}{2} x_j^2 + b_j x_j + c_j\right) + \frac{a}{2} x_n + \frac{b}{x_n}$ and show that, under suitable parameters, $f$ is globally defined and $\lambda_F \le 0$ (with special Ricci-flat fiber cases when $a=0$). The results yield complete criteria for constructing Einstein warped products and provide explicit models in the hyperbolic-base setting, contributing to the broader classification of Einstein spaces with warped product structures.
Abstract
We study Einstein riemannian manifolds endowed with a warped product structure. We focus on the case in which both the base and the fiber are Einstein manifolds and establish necessary and sufficient conditions for the warped product itself to be an Einstein manifold. Moreover, we explicitly determine the warping function when the base is the hyperbolic space.