  In this paper, we give an expression and some estimates of the curvature
tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau
threefold. The symmetricity of the Yukawa coupling is also studied. In the last
section of this paper, an extra restriction of the limiting Hodge structure for
the degeneration of Calabi-Yau threefolds is given.
