Abstract: In joint work with Hao Fang, I introduce and prove the existence of metrics on complex surfaces with split tangent bundle. These metrics are analogous to Calabi-Yau metrics, as they flatten certain holomorphically trivial line bundles adapted to the geometric structure, in this case the splitting. First, we will review the Calabi-Yau theorem in the Kahler setting and some issues with generalizing it to non-Kahler manifolds. Then, I will discuss some machinery — introduced by Streets -- that makes it possible to reduce this problem to the study of a family of non-concave full-nonlinear elliptic PDE. Finally, I will show that these PDE are smoothly solvable and draw some parallels to the twisted Monge-Ampere equation.