Coordinate tranformations on an embedded surface of in 3 space

[lavinia]

Given a surface of positive curvature embedded in R^3 choose coordinate charts around each non-umbilic point so that the cross terms in both the first and second fundamental forms are zero.

These are coordinates where the tangents to the coordinate axes point in the direction of the principal curvatures.

For two overlapping such charts (u,v) and (x,y) what are the properties of the coordinate transformation between them?

 

[lavinia]

It seems that the coordinate transformation can be cosen to be either the identity, a reflection, or pure negation depending on whether one chooses to locally negate one or both coordinate vector fields. These transformations, if chosen correctly ,then can be taken to be conformal.

Take a regular ellipsoid with 4 umbilics. At the umbilics there are no preferred directions of curvature so the coordinates can not be extended to them. What about the conformal structure?

The coefficient of the Beltrami differential is (e-g)/(sqrt(e) + sqrt(g))^2 - I think, where e and g are the non- zero components of the second fundamental form. This expression becomes zero at the umbilics and gives a conformal structure on the whole ellipsoid.

This shows that the conformal structure depends only on the second fundamental form.

What is the coordinate chart around the umbilics that extends the conformal structure?

 

[lavinia]

there is a mistake here.