A question about Gauss-Bonnet theorem on real and complex manifolds Let $(M,g)$ be a $2$-dimension complex manifold then one can apply Gauss-Bonnet theorem an get some results. On the other hand, this manifold is a real $4$-dimension Riemannian manifold which satisfies on Gauss-Bonnet theorem!! Is my conclusion correct? Thanks

There's only one (Chern) Gauss-Bonnet Theorem for even-dimensional compact Riemannian manifolds (perhaps with boundary). You will integrate what is called the Pfaffian of the curvature $2$-form matrix of the $4$-dimensional Riemannian manifold. On the other hand, you can consider the integral of the second Chern form of the complex $2$-dimensional manifold with its hermitian metric. Both of these will give you (up to constants) the Euler characteristic of the manifold.