Positive Gauss Curvature Metric: Calculation & Analysis

[lavinia]

Given is a surface embedded in Euclidean 3 space whose Gauss curvature is everywhere positive.

Its metric is <Xu,Xu> = E <Xu,Xv> = F <Xv,Xv> = G for an arbitrary coordinate neighborhood on the surface.

The principal curvatures determine a new metric. In principal coordinates this new metric is

<Xu,Xu> = k1$$^{2}$$E <Xu,Xv> = 0 <XvXv> = k2$$^{2}$$G

Is this also a metric of positive Gauss curvature?

 

[lavinia]

I think the answer is yes. Here is the argument.

The new metric is induced from the standard metric on the unit 2 sphere under the Gauss map.

If Xu is a principal direction with norm E^.5 then dG(Xu) = k1Xu has norm k1E^.5.

Thus if S,T is a orthonormal basis on the unit sphere that is in the image under the Gauss map of two principal directions then their pull backs are dual to the vectors

1/(k1E^.5) Xu and 1/(k2G)^.5Xv and Xu and Xv now have the lengths k1E^.5 and k2G^.5

It follows that the curvature of the new metric is identically equal to 1. Yes?