  We prove that the period maps from the Torelli space and the moduli space
with level $m$ structure of Calabi-Yau type manifolds to the corresponding
period domain of polarized Hodge structures are injective. The proof is based
on the construction of holomorphic affine structure on the Teichm\"uller space
and the construction of the completion space of the Torelli space with respect
to the Hodge metric.
