A basic task in geometric algorithms is finding *mappings* between surfaces, which can be used to transfer data from one place to another. A particularly nice class of algorithms (both from a mathematical and computational perspective) are *conformal* mappings, which preserve angles between vectors, and are generally very well-behaved. In this lecture we’ll take a look at smooth characterizations of conformal maps, which will ultimately inspire the way we talk about conformal maps in the discrete/computational setting.

The video covering both today and Thursday’s lecture (on discrete aspects of conformal maps) can be found here.

I have two questions that may or may not directly linked to this lecture:

1. Are $X$ and $\mathcal{J}X$ vector fields on the manifold or its coordinate chart?

2. When we integrate a 2- form $\omega$ on M. Since it’s defined as $\int_M{\omega(X,Y)}$, how do we know it’s irrelevant to the choice of X and Y. What about replacing $X$ and $Y$ with $X$ and $\mathcal{J}X$?