In Siu's Lecture note: Lectures on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics, he shows the $C^0$ and $C^2$ estimates of the complex Monge-Ampere equation on Riemannian manifold; and then uses Evans-Krylov's arguments to show Holder estimates for the second derivatives. But Chapter two §4 (p.100), he says "Let $u$ be a real-valued function on an open subset of $\mathbb{C}^m$".
My question is why it suffices to show the $C^{2,\alpha}$ estimate in euclidean space instead of the Riemannian manifold.