Translation Surfaces in Lie Groups with Constant Gaussian Curvature

Let G be an n-dimensional \((n\ge 3)\) $(n\ge 3)$ Lie group with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in G can be expressed as the product of two curves, then it must be flat. In particular, we can essentially characterize all such surfaces locally in the three-dimensional case.