Minkowski spacetime and non-Ricci-flat compactification in heterotic supergravity

We compactify the ten-dimensional spacetime in heterotic supergravity leaving four-dimensional Minkowski spacetime. We search for nonsupersymmetric, non-Ricci-flat solutions of the equations of motion with the quadratic curvature term. By assuming that the extradimensional spaces are products of 2-manifolds, three types of solutions are found. They are $S^2\times T^2 \times H^2/\Gamma$, $S^2\times H^2/\Gamma \times H^2/\Gamma$, and $S^2\times S^2 \times H^2/\Gamma$, where $H^2/\Gamma$ denotes a compact hyperbolic manifold. The metrics can be written explicitly, and they can be applied to phenomenology.