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 this be generalized?

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?