New proofs of the Torelli theorems for Riemann surfaces

TL;DR

This paper provides new differential geometric proofs of the local and global Torelli theorems for Riemann surfaces using explicit deformation formulas and Kuranishi coordinates.

Contribution

It introduces explicit expressions for the period map and offers novel proofs of the Torelli theorems based on differential geometry.

Findings

Explicit formulas for the period map derived

New differential geometric proofs of Torelli theorems presented

Both local and global Torelli theorems proved using these methods

Abstract

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