# Conformal structure determined by principal curvatures

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?

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What do you do at umbilic points? –  Will Jagy Apr 16 '11 at 19:59
In other words you want to use second fundamental form as a metric tensor and see what happens. I do not know nice geometric meaning for the obtained space and would be surprised if there is a nice one. For conformal structure --- there is only one conformal structrure on the sphere... –  Anton Petrunin Apr 16 '11 at 21:04
This metric is sometimes used in constructing an approximating inscribed polyhedron for the surface. There are some optimality results, e.g. K. Clarkson (2006) "Building triangulations using epsilon nets" and the works of P. M. Gruber cited therin. –  Ramsay Apr 17 '11 at 1:04
It is worth noting that everything works equally well in higher dimensions. The second fundamental form of a strongly convex hypersurface in Euclidean space determines a second Riemannian metric on the hypersurface, in addition to the one induced by the Euclidean inner product. If you forget the Euclidean structure and use only the affine structure of the ambient space, then the second fundamental form is still well-defined up to a scalar factor and you get a conformal structure. Both of these have been studied extensively. The latter is used to derive local invariants of a Finsler manifold. –  Deane Yang Apr 18 '11 at 3:25

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.

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- Unless I am not thinking straight, it seem that the curvature of the pull back metric from the sphere is constant. - The other metric I could not show has positive curvature. If there were an infinite sequence of metrics produced by repeated choice of this principal curvature metric then would there have to be a limiting geometry? On the surface (so to speak) this seems impossible since it would have to be the constant curvature metric on the sphere. So maybe after a while the metric can not be embedded in 3 space. –  marc Apr 30 '11 at 20:18
When defined as a tensor taking values in the normal bundle the second fundamental form of a hypersurface in $\mathbb{R}^{n+1}$ depends only on the affine structure on the ambient space, not on its metric structure. On the locus where it is non-degenerate it determines a conformal structure which has been used in studying the affine geometry of the hypersurface at least since the 1930's (see the second volume of Blaschke's book Vorlesungen über Differentialgeometrie entitled Affine Differentialgeometrie). If one restricts attention to those aspects of the geometry of the hypersurface preserved by unimodular affine transformations, then a particular representative, called the Blaschke or equiaffine metric, is determined. With respect to a Euclidean unit normal, it is the second fundamental form multiplied by the $-1/(n+2)$ power of the Gaussian curvature. That is, if the hypersurface is written locally as the graph of $f$ then the metric is given in coordinates $|\det f_{ij}|^{-1/(n+2)}f_{ij}$, where $f_{ij} = \tfrac{\partial^{2}f}{\partial x^{i} \partial x^{j}}$. When the second fundamental form is locally strictly convex, this metric is Riemannian, and much use of it has been made in the study of such hypersurfaces. To my taste the best introduction to these matters is provided by two papers of Calabi: Complete affine hyperspheres. I (there is no sequel) and Hypersurfaces with maximal affinely invariant area.
In the two-dimensional case the conformal structure determines a complex structure, and it helps to use it. For this two basic references are Changping Wang's Some examples of complete hyperbolic affine $2$-spheres in $\mathbb{R}^{3}$ and Calabi's Convex affine maximal surfaces.