Conformal structure determined by principal curvatures

Question

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?

Answer 1

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.

Answer 2

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.

When the hypersurface is non-degenerate there can be defined the affine normal, which is a vector field transverse to the hypersurface that transforms equivariantly under the action of unimodular affine transformations on the ambient space. Via the splitting induced by this normal the flat affine connection on the ambient space induces an affine connection on the hypersurface. The difference of this affine connection with the Levi-Civita connection of the equiaffine metric is a tensor, which if its contravariant index is lowered using the metric, is completely symmetric. This completely symmetric covariant tensor three tensor (the Pick form) and the equiaffine metric encode the geometry of the hypersurface. Such a pair can be considered in the abstract, and the local obstructions to realizing it as that induced on a non-degenerate hypersurface are given in terms of the curvature of the affine connection and the Pick form, and, if I remember right, can be found in the textbook Affine differential geometry of Nomizu and Sasaki. Where such obstructions come from can be seen as follows. The map assigning to a point of the hypersurface the annihilator of its tangent space takes values in the projectivization of the vector space dual to the ambient space; this map is an immersion if and only if the hypersurface is non-degenerate. In this case the pullback via this map of the flat projective structure induces a flat projective structure on the hypersurface. A particular connection representative of this projective structure is distinguished by the requirement that its difference with the Levi-Civita connection of the equiaffine metric be the negative of the Pick form. Then for an arbitrary pair comprising a metric and a completely symmetric covariant three tensor to be that induced on an immersed hypersurface, a necessary condition is that the connection formed by adding to the Levi-Civita connection the negative of the three tensor (with an index raised) be projectively flat.

As far as I am aware the question of when such a pair can be realized globally via a proper embedding has not been studied other than in some particular cases, determined by conditions on the affine shape operator of the putative embedding, conditions which can be stated solely in terms of the Ricci curvature of the metric and the Pick form. Much of it boils down to controlling the size of the Pick form. For instance, the heart of Calabi's paper on complete affine hyperspheres is a differential inequality for the Laplacian (in the equiaffine metric) of the norm of the Pick form. In this regard, in addition to the aforementioned papers of Calabi, N. Trudinger and X.-J. Wang's survey The Monge-Ampère equation and its geometric applications and J. Loftin's Survey on affine spheres are good starting points.

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.