9
This weird formula for closed 1-forms on a Riemann surface
Post Body
I keep stumbling on this equality in articles, but I have no idea where it comes from. It's always quoted as "well-known" except for a reference to a textbook I have no access to.
Take a genus g compact Riemann surface [;\Sigma;]
, fix a symplectic basis of homology cycles [;a_i,b_i;]
and open it up in a fundamental polygon as in this picture. Then, if [; \theta, \eta ;]
are closed 1-forms, this holds apparently:
[; \int_\Sigma \theta \wedge \eta = \sum_{i=1}^g \left( \int_{a_i} \theta \int_{b_i} \eta - \int_{a_i} \eta \int_{b_i} \theta \right) ;]
I cannot manage prove exactly this.
Author
Account Strength
100%
Account Age
10 years
Verified Email
Yes
Verified Flair
No
Total Karma
89,343
Link Karma
3,504
Comment Karma
85,784
Profile updated: 20 hours ago
Posts updated: 9 months ago
Subreddit
Post Details
We try to extract some basic information from the post title. This is not
always successful or accurate, please use your best judgement and compare
these values to the post title and body for confirmation.
- Posted
- 9 years ago
- Reddit URL
- View post on reddit.com
- External URL
- reddit.com/r/puremathema...