Flat 3-manifolds with diagonal metrics and applications to warped products

We provide necessary and sufficient conditions for a $3$-dimensional submanifold of ${\mathbb{R}}^3$ endowed with a diagonal metric to be flat. As applications, we characterize the flat manifolds of warped product-type, more precisely, the warped, biwarped, sequential warped, and doubly warped product manifolds, and we state the corresponding nonexistence results.