Tori Embedded in S3 with Dense Asymptotic Lines

In this paper are given examples of tori T^2 embedded in S^3 with all their asymptotic lines dense.


Introduction
Let α : M → S 3 be an immersion of class C r , r ≥ 3, of a smooth, compact and oriented two-dimensional manifold M into the three dimensional sphere S 3 endowed with the canonical inner product < ., . > of R 4 .
Here, N α is the positive unit normal of the immersion α and < N α , α >= 0.
Through every point p of the hyperbolic region H α of the immersion α, characterized by the condition that the extrinsic Gaussian Curvature K ext = det(DN α ) is negative, pass two transverse asymptotic lines of α, tangent to the two asymptotic directions through p. Assuming r ≥ 3 this follows from the usual existence and uniqueness theorems on Ordinary Differential Equations. In fact, on H α the local line fields are defined by the kernels L α,1 , L α,2 of the smooth one-forms ω α,1 , ω α,2 which locally split II α as the product of ω α,1 and ω α,2 .
The forms ω α,i are locally defined up to a non vanishing factor and a permutation of their indices. Therefore, their kernels and integral foliations are locally well defined only up to a permutation of their indices.
Under the orientability hypothesis imposed on M, it is possible to globalize, to the whole H α , the definition of the line fields L α,1 , L α,2 and of the choice of an ordering between them, as established in (Garcia and Sotomayor 1997) and (Garcia et al. 1999).
These two line fields, called the asymptotic line fields of α, are of class C r−2 on H α ; they are distinctly defined together with the ordering between them given by the subindexes {1, 2} which define their orientation ordering: "1" for the first asymptotic line field L α,1 , "2" for the second asymptotic line field L α,2 .
The asymptotic foliations of α are the integral foliations A α,1 of L α,1 and A α,2 of L α,2 ; they fill out the hyperbolic region H α .
In a local chart (u, v) the asymptotic directions of an immersion α are defined by the implicit differential equation In S 3 , with the second fundamental form relative to the normal vector N = α ∧ α u ∧ α v , it follows that: There is a considerable difference between the cases of surfaces in the Euclidean and in the Spherical spaces. In R 3 the asymptotic lines are never globally defined for immersions of compact, oriented surfaces. This is due to the fact that in these surfaces there are always elliptic points, at which K ext > 0 (Spivak 1999, Vol. III, chapter 2, pg. 64).
In (Garcia and Sotomayor 1997) and (Garcia et al. 1999) ideas coming from the Qualitative Theory of Differential Equations and Dynamical Systems such as Structural Stability and Recurrence were introduced into the subject of Asymptotic Lines. Other differential equations of Classical Geometry have been considered in (Gutierrez and Sotomayor 1991); a recent survey can be found in (Garcia and Sotomayor 2008a).
The interest on the study of foliations with dense leaves goes back to Poincaré, Birkhoff, Denjoy, Peixoto, among others.
In this paper is given an example of an embedded torus (deformation of the Clifford torus) with both asymptotic foliations having all their leaves dense.

Preliminary Calculations
In this section will be obtained the variational equations of a quadratic differential equation to be applied in the analysis in Section 3. Proposition 1. Consider a one parameter family of quadratic differential equations of the form (1) Then the following variational equations holds: (2) Proof. Differentiation with respect to ǫ of (1) written as taking into account that a v = ∂a ∂v , a ǫ = ∂a ∂ǫ , a ǫu = a uǫ = ∂ 2 a ∂ǫ∂u = ∂ 2 a ∂u∂ǫ , leads to: Analogous notation for b = b(u, v(u, v 0 , ǫ), ǫ), c = c(u, v(u, v 0 , ǫ), ǫ) and for the solution v(u, v 0 , ǫ).
Evaluation of equation (3) at ǫ = 0 results in: Differentiating twice the equation (1) and evaluating at ǫ = 0 leads to: Similar calculation gives the variational equations for u ǫ and u ǫǫ . This ends the proof.

Consider the Clifford torus
Proposition 2. The asymptotic lines on the Clifford torus in the coordinates given by equation (5) are given by dudv = 0, that is, the asymptotic lines are the coordinate curves (Villarceau circles). See Fig. 1.

Figure 1. Torus and Villarceau circles
Proof. The coefficients of the first fundamental form I = Edu 2 + 2F dudv + Gdv 2 and the second fundamental form II = edu 2 +2f dudv+gdv 2 of C with respect to the normal vector field N = C ∧ C u ∧ C v are given by: Therefore the asymptotic lines are defined by dudv = 0 and so they are the coordinate curves. Fig. 1 is the image of the Clifford torus by a stereographic projection of S 3 to R 3 .
Theorem 1. There are embeddings α : T 2 → S 3 such that all leaves of both asymptotic foliations, A α,1 and A α,2 , are dense in T. See Fig. 2 Figure 2. Stereographic projection of a deformation of a Clifford torus with ǫ = 2/3. (u, v) be the unit normal vector to the Clifford torus.
Therefore, the integration of the linear differential equations (9) leads to: Taking h(u, v) = sin 2 (2v − 2u), it results from equation (7) that: In fact, from the definition of h it follows that: So, a careful calculation shows that equation (12) follows from equation (7).

Concluding Comments
In this paper it was shown that there exist embeddings of the torus in S 3 with both asymptotic foliations having all their leaves dense.
In (Garcia and Sotomayor 2008b) is given an example of an embedded torus in R 3 with both principal foliations having all their leaves dense.
The technique used here is based on the second order perturbation of differential equations.
It is worth mentioning that the consideration of only the first variational equation was was technically insufficient to achieve the results of this paper.