  This is a continuation of our previous paper [14]. In [14], we introduced the
first Aeppli-Chern class on compact complex manifolds, and proved that the
$(1,1)$ curvature form of the Levi-Civita connection represents the first
Aeppli-Chern class which is a natural link between Riemannian geometry and
complex geometry. In this paper, we study the geometry of compact complex
manifolds with Levi-Civita Ricci-flat metrics and classify minimal complex
surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that
minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are K\"ahler
Calabi-Yau surfaces and Hopf surfaces.
