Three-generation solutions of equations of motion in heterotic supergravity

We study the generation number of massless fermions in compactifications recently found in heterotic supergravity. The internal spaces are products of two-dimensional spaces of constant curvature, and the standard embedding is not assumed. The generation number is constrained by the equations of motion and the Bianchi identity. In the case that the Euler characteristics of the three internal submanifolds are $(2,-2,-2)$, the generation number is three or less. We also show that three-generation solutions exist for arbitrary Euler characteristics of the negatively curved 2-manifolds, and we present some explicit solutions.