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In this paper, by using the Kuranishi coordinates on the Teichm\"uller spaceand the explicit deformation formula of holomorphic one-forms on Riemannsurface, we give an explicit expression of the period map and derive newdifferential geometric proofs of the Torelli theorems, both local and global,for Riemann surfaces.

New proofs of the Torelli theorems for Riemann surfaces