On a smooth surface of positive Gauss curvature that is embedded in Euclidean 3 space the principal curvatures seem to define a smooth field of ellipses whose major and minor axes are lined up with the directions of curvature and whose ratio of lengths is k1/k2, the ratio of the principal curvatures. 


My first question is whether there is true or whether there is a flaw in this observation.


If true, the embedding determines a conformal structure and I assume that the associated Riemannian metric is determined by the usual equations


1 + |u|/ 1 - |u| = k1/k2 or |u| = k1 - k2/ k1 + k2 with arg(u) = 0. So ds^2 = (dz + udz\_)^2


My second question is whether this is correct.


Finally, if this construction works what can be said about the possible conformal structures and Riemannian geometries that can be obtained in this way from embeddings of surfaces of positive curvature in 3 space? Can the new metric be realized in 3 space? 


Can this be generalized?



Conformal structure determined by principal curvatures
When you define an actual ellipse field, you get more than a conformal structure, you get a Riemannian metric.


There are two natrual and reciprocal ways to define an ellipse field from the principle curvatures of a smooth convex surface
such that the ratio of major and minor axes is $k\_1/k\_2$.
If $x\_1$ and $x\_2$ are the principle directions with principle curvatures $k\_1$ and $k\_2$, then one natural definition would be to use the equation $(k\_1 x\_1)^2 + (k\_2 x\_2)^2 =1$.
In this case the axis of the ellipse in the $x\_i$ direction has length $1/k\_i$.
The equation says that the first derivative of the Gauss map, applied to the tangent vector $(x\_1, x\_2)$, has norm 1. The metric is thus the pullback of the metric on the unit sphere by the Gauss map. (The Gauss map is the map surface $\to S^2$ takes a point on the surface to the unit normal vector at that point)


If you prefer the other ellipse field $(x\_1/k\_1)^2 + (x\_2/k\_2)^2 = 1$, you can think of this as going in the opposite direction from the round metric induced by the Gauss map, by an equal amount. 


More specifically, the space of positive definite quadratic forms on a 2-dimensional vector space is a homogeneous space, since $GL(2,\mathbb R)$ acts transitively via change of basis: in matrix form, an element $A \in GL(2,\mathbb R)$ sends a quadratic form $Q$ to $A^t Q A$.
The stabilizer of any particular quadratic form is isomorphic to $O(2)$. There is a natural invariant Riemannian metric on this space making it isometric to $H^2 \times \mathbb E^1$, the hyperbolic plane cross a line. The line direction is proportional to the log of the determinant of the matrix for the form. (Choose your preferred constant). The two
metrics given by diagonal matrices with entries $(k\_1^2, k\_2^2)$ or $(k\_1^{-2}, k\_2^{-2})$ are on a straight line segment in this geometry with midpoint the identity matrix. 


So, you can think of the second definition as a political cartoon of the surface, with features exaggerated to make them twice as prominent, in a certain sense as above. For surfaces that are nearly spherical in the $C^2$ topology, the new metric will still have positive curvature, and by Alexandrov's theorem it can still be embedded isometrically in $\mathbb E^3$ as a convex surface. The embedding is unique up to isometry. This will amount to making the bulges about twice as big, and the hollows about twice as deep.


