Constant curvature hypersurfaces of cylinders over space forms

We classify the hypersurfaces of dimension n >= 3 with constant sectional curvature in the product spaces R^k x Sⁿ⁻ᵏ⁺¹ and R^k x Hⁿ⁻ᵏ⁺¹, for 2 <= k <= n-1. Our results provide a complete description of these hypersurfaces and extend previous classifications of constant curvature submanifolds in product spaces of space forms.