Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New proofs of the Torelli theorems for Riemann surfaces   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.

25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex