Doubly warped product Einstein metrics on spheres

Abstract: We present a simple computer-assisted procedure to construct $SO(d_1+1)\times SO(d_2+1)$-invariant cohomogeneity one Einstein metrics, and use it to recover known Einstein metrics on $S<sup>{10}$ and $S<sup>{12}$, as well as find new ones on $S<sup>{11}$, $S<sup>{12}$, $S<sup>{13}$ and $S^7\times S<sup>3$.