Slight generalization of the Local Normal Form

 

The Local Normal Form states that if F:X\to Y is a holomorphic map at p\in X, which is not constant, then there is a unique integer m\geq 1 which satisfies the following property: for every chart \phi_2: U_2\to V_2 on Y centered at F(p), there exists a chart \phi_1: U_i\to V_1 on X centered at p such that \phi_2(F(\phi_1^{-1}(z)))=z^m.

The way I understand the proof, I think we can extend it to say that for any chart U_1\to V_1 on X centered at p, then there exists a chart \phi_2: U_2\to V_2 on Y centered at F(p) such that \phi_2(F(\phi_1^{-1}(z)))=z^m. The only added condition is that \phi_2: U_2\to V_2 is non-zero except at F(p).

Proving this would be quite easy, and is left as an exercise.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics.

One thought on “Slight generalization of the Local Normal Form

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: