Kodaira dimensions and hyperbolicity of nonpositively curved compact Kähler manifolds

In this article, we prove that a compact Kähler manifold M ⁿ with real analytic metric and with nonpositive sectional curvature must have its Kodaira dimension, its Ricci rank and the codimension of its Euclidean de Rham factor all equal to each other. In particular, M ⁿ is of general type if and only if it is without flat de Rham factor. By using a result of Lu and Yau, we also prove that for a compact Kähler surface M ² with nonpositive sectional curvature, if M ² is of general type, then it is Kobayashi hyperbolic.