Symmetric solutions of Einstein's equations in higher dimensions

M. Jakimowicz and J. Tafel

We consider (n+N+1)- dimensional metrics invariant under the rotation group SO(n+1) acting on n-dimensional spheres. We first reduce the Einstein equations to equations in N+1 dimensions with a scalar field s and an exponential potential. The latter equations are studied under the assumption that the normal field to surfaces s=const is geodetic and the induced metric of the surface satisfies the Einstein condition. We obtain a closed system of two ordinary differential equations (they correspond to the Friedmann equations in cosmology) and algebro-geometric conditions on the metric of the surfaces. We present examples of multidimensional vacuum metrics obtained by this method. Among them there are solutions which generalize the 5-dimensional Gross-Perry metric and new solutions which belong to the Kundt class.