  In this paper, we define and study the Weil-Petersson geometry. Under the
framework of the Weil-Petersson geometry, we study the Weil-Petersson metric
and the Hodge metric. Among the other results, we represent the Hodge metric in
terms of the Weil-Petersson metric and the Ricci curvature of the
Weil-Petersson metric for Calabi-Yau fourfold moduli. We also prove that the
Hodge volume of the moduli space is finite. Finally, we proved that the
curvature of the Hodge metric is bounded if the Hodge metric is complete and
the dimension of the moduli space is 1.
