A metric property of umbilic points

In the space $\mathbb U^4$ of cubic forms of surfaces, regarded as a $G$-space and endowed with a natural invariant metric, the ratio of the volumes of those representing umbilic points with negative to those with positive indexes is evaluated in terms of the asymmetry of the metric, defined here. A connection of this ratio with that reported by Berry and Hannay (1977) in the domain of Statistical Physics, is discussed.


Umbilic Points, Invariant Metrics and Volume Ratios
At an umbilic point p of an oriented C 3 surface S embedded in an oriented Euclidean 3-space R 3 the principal curvatures coincide. In a neighborhood of such point, S can be written in a Monge chart as the graph z = h(x, y) of a function of the form h(x, y) = k 2 (x 2 + y 2 ) + 1 6 (ax 3 + 3bx 2 y + 3b ′ xy 2 + a ′ y 3 ) + o((x 2 + y 2 ) 3/2 ).
The frame (x, y; z) is positive and adapted to S at p. This means that the plane orthonormal frame (x, y) is attached to the tangent plane, positively oriented, and the z-axis is along the unit positive normal to S at p.
Any other such presentation as the graph Z = H(X, Y ) of a function differs by a rotation x = cos θX − sin θY, y = sin θX + cos θY, z = Z, linking the positively oriented frames (X, Y ; Z) and (x, y; z), adapted to the surface at p.
The functions are related by H(X, Y ) = h(x, y); substitution leads to K = k A = a cos 3 θ + 3b cos 2 θ sin θ + 3b ′ sin 2 θ cos θ + a ′ sin 3 θ Thus, the group O(2) of rotations in the plane acts linearly, to the right, on the four dimensional space of real cubic forms Denote by Ω(θ) the matrix of the linear transformation in R 4 , corresponding to the frame rotation by an angle θ.
From equation 2, get The space U 4 of umbilic intrinsic cubic forms on surfaces is defined as the G-Space R 4 , endowed with the above action of the group G = O(2).
The quadratic form Thus it is defined on U 4 . It characterizes the transversal umbilic points, as those with T (u) = 0. It is well known that the Index I(u) of a transversal umbilic u is I(u) = 1 2 sign(T (u)). See [2] and [13,15] where the identification of T = 0 with the transversality to the manifold of umbilic 2-jets is made.
The index of an isolated umbilic counts the number of turns made by a principal direction at a point of the surface that makes a small circuit around the umbilic, [21] and [20].
According to [21] and [22], the differential equation of principal lines around p in this chart is defined as a variety in the Projective Bundle. In the chart (x, y, [dx : dy]), the variety is given by the equation: where the functions L, M and N are: where β > 0, and β( 2 3 α + 1 3 β) − α 2 > 0, which gives the positivity of Q.
The criterion for the positivity of a symmetric matrix, consisting in that of all the principal minors, is proved in Gantmacher [5], Chap. X, pg. 306. Notice that for Q in (6) the positivity of the second and third principal minors imply that of the other two.
Remark 1. Another way to obtain the expression of Q in (6) consists in projecting the 10 dimensional space M 10 of 4 × 4 symmetric matrices M via the averaging A along the orbits of Ω(θ): Denoting by m ij the entries of the symmetric matrix M , integration in expression (7) gives : for the invariant symmetric matrix A(M ). The other entries of Q in (6) are also corroborated by integration in (8).
Proof. The invariance of the planes is straightforward.
The second and third items follow from a straightforward calculation.
Although other possibilities exist, in this work the forms r 1 and r 2 will be used as a reference.
The asymmetry of q is defined by the ratio σ(q) = m 2 /m 1 .
Clearly σ(q) = m 2 /m 1 ranges over all positive reals. An expression for it in terms of α, β has been given in remark 2.

Theorem 2. Let T be the quadratic form in equation 4, giving the index of transversal umbilic points.
Relative to unit ball B(1, q) = {q(u) ≤ 1} of any invariant metric q in U 4 , the ratio of the volume V − of the cone C − , where T is negative, to that of the volume V + of the cone C + , where T is positive, is given by 9(σ(q)) 2 , where σ(q) is the asymmetry of q, as in definition 1 and remark 2.
Therefore, in terms of q 1 , q 2 , The proof consists in computing the volume V − of the solid torus cone and divide it by the volume V + of the solid torus cone Let v i1 , v i2 be an orthonormal basis of U i , i = 1, 2, relative to q i , so that they form a positive orthonormal frame, relative to q, on U 4 .
In q-orthonormal coordinates (x, y, z, w) relative to the frame v i1 , v i2 , i = 1, 2, it follows that Let x = r cos θ, y = r sin θ and z = R cos γ, The element of volume dV in the metric q is given by dxdydzdw Therefore, dV = rRdrdRdθdγ and rRdrdR.
Take tan β 0 = m 1 3m 2 , the volume of the solid torus cone C + is given by , Analogously, the volume of the solid torus cone C − is equal to

Remark 2.
In terms of Q, as in equation 6, σ(q) is calculated as follows: Proof. In fact, by the uniqueness of the simultaneous diagonalization of the quadratic forms q and T , see [5] pg. 314, equation 9 implies that the eigenvalues of matrix M T , of T , relative to Q, the matrix of q, are 72( m 1 3 ) 2 and −72m 2 2 . Separate direct calculation of these relative eigenvalues, which are those of the matrix M T Q −1 , gives 1 2(3α+β) and − 3 2(β−α) . Equating the ratios of the eigenvalues in both calculations gives 9(m2/m1) 2 =3 3α+β β−α , which amounts to equation (10).

At the Crossroads of Geometry and Global Analysis
The Geometric local properties of umbilic points, regarded as singularities, have been studied focusing the three following main aspects: i) Topological, related to the Index sign of the principal line fields around the umbilic. ii) Focal, describing the patterns, Hyperbolic Elliptic, of normal rays envelopes.
This aspect is related to Geometric Optics, Catastrophe Theory and Lagrangian Geometry. See [23,24]. iii) Darbouxian, which counts the number of principal lines separatrices approaching the umbilic, (D 1 , D 2 or D 3 ) and, more generally, describes locally the foliations by principal lines. These aspects and their different types are discriminated and analyzed in terms of suitable algebraic conditions in the G-space U 4 .
A coherent differential geometric and topological picture of the set morphology and inclusion relationships between the different sorts of umbilic types has been established by Porteous, see [19], and previous reference quoted there. See also Zeeman [25], for the focal aspect, and Darboux [4], Sotomayor-Gutierrez [13,14,15] and Bruce-Fidal [2], for the Darbouxian types.
The globalization to the whole surface of the local analysis of Darboux, in the context of Structural Stability and Genericity of principal foliations, was carried out in [13,14,15].
An additional extension led Gutierrez, Garcia, Sotomayor and others, to expand the study of umbilic points and also principal foliations to surfaces and hypersurfaces in R 4 . See [6], [16], [12].
There remain deep open problems related to structure of principal foliations around isolated umbilic points in smooth surfaces, in the non-transversal case. See Mello-Sotomayor [18], Smyth-Xavier [20] and Ivanov [17].

Umbilic Points in Random Surfaces
On the domain of Statistical Physics, but still connected to Geometry and Topology, Berry and Hannay [1] carried out a quantitative statistical study of the proportions in which the different types of umbilic types are distributed in random surfaces, such as those modeling an ocean or a lake. An issue here is to study how the presence of umbilic points in a random surface influences the reflection on it of electromagnetic short waves. Although this work is more related to the focal interpretation of umbilic points, it considers explicitly also their Darbouxian and Index aspects.
This paper is the outcome of an initial attempt to provide a mathematical formulation and a proof, in the tradition of Geometry and Classical Analysis, that could correspond to the conclusions of Berry and Hannay, [1], reported in the tradition of Statistical Physics.
Theorem 2 suggests a disagreement with the report of the calculations in [1] which claim that the statistical ratio is always 1, disregarding of the statistic anisotropy present in the evaluation. The asymmetry of the invariant metric, used to make evaluations in this work, may be considered as a geometric counterpart for the statistic anisotropy.
Considering only the local aspect of surfaces at umbilic points, this discrepancy may be due to the fact that in the calculations made in [1], the cubic forms are regarded as vectors in R 4 , with a fixed frame, and not as elements of the G-space U 4 . The effect of this is that the same umbilic on a surface is counted multiple times, one for each rotated frame.