In this paper, we introduce the first Aeppli-Chern class for complex
manifolds and show that the $(1,1)$- component of the curvature
$2$-form of the Levi-Civita connection on the anti-canonical line
bundle represents this class. We systematically investigate the
relationship between a variety of Ricci curvatures on Hermitian
manifolds and the background Riemannian manifolds. Moreover, we
study non-K\"ahler Calabi-Yau manifolds by using the first
Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In
particular, we construct explicit Levi-Civita Ricci-flat metrics on
Hopf manifolds $\S^{2n-1}\times \S^1$. We also construct a smooth
family of Gauduchon metrics on a compact Hermitian manifold such
that the metrics are in the same first Aeppli-Chern class, and their
first Chern-Ricci curvatures are the same and nonnegative, but their
Riemannian scalar curvatures are constant and vary smoothly between
negative infinity and a positive number. In particular, it shows
that
Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar
curvature.