Tori embedded in S³ with dense asymptotic lines

Abstracts

In this paper are given examples of tori T² embedded in S³ with all their asymptotic lines dense.

asymptotic lines; recurrence; Clifford torus; variational equation

Neste artigo são dados exemplos de toros T² mergulhados em S³ com todas as suas linhas assintóticas densas.

linhas assintóticas; recorrências; toro de Clifford; equação variacional

MATHEMATICAL SCIENCES

Tori embedded in S³ with dense asymptotic lines

Ronaldo Garcia^I; Jorge Sotomayor^II

^IInstituto de Matemática e Estatística, Universidade Federal de Goiás Caixa Postal 131, 74001-970 Goiânia, GO, Brasil

^IIInstituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1.010 Cidade Universitária, 05508-090 São Paulo, SP, Brasil

^{Correspondence to}

ABSTRACT

In this paper are given examples of tori T² embedded in S³ with all their asymptotic lines dense.

Key words: asymptotic lines, recurrence, Clifford torus, variational equation.

RESUMO

Neste artigo são dados exemplos de toros T² mergulhados em S³ com todas as suas linhas assintóticas densas.

Palavras-chave: linhas assintóticas, recorrências, toro de Clifford, equação variacional.

1 INTRODUCTION

Let α: → ³ be an immersion of class C^r, r > 3, of a smooth, compact and oriented two-dimensional manifold into the three dimensional sphere ³ endowed with the canonical inner product 〈·,·〉 of ⁴.

The Fundamental Forms of α at a point p of are the symmetric bilinear forms on _p defined as follows (Spivak 1999):

I_α (p; v, w) = 〈Dα(p; v), Dα(p; w)〉,

II_α (p; v, w) = 〈-DN_α(p; v), Dα(p; w)〉.

Here, N_α is the positive unit normal of the immersion a and 〈N_α, α〉 = 0.

Through every point p of the hyperbolic region _α of the immersion a, characterized by the condition that the extrinsic Gaussian Curvature _ext = det(D N_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 _α the local line fields are defined by the kernels _{α ,1}, _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 , it is possible to globalize, to the whole _α, the definition of the line fields _α,1, _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 _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 _α,1, "2" for the second asymptotic line field

_α,2.

The asymptotic foliations of a are the integral foliations _α,1 of _α,1 and _α,2 of _α,2; they fill out the hyperbolic region _α.

In a local chart (u, v) the asymptotic directions of an immersion a are defined by the implicit differential equation

In

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

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 _ext > 0 (Spivak 1999, Vol. III, chapter 2, pg. 64).

The study of asymptotic lines on surfaces of ³ and ³ is a classical subject of Differential Geometry. See (do Carmo 1976, chapter 3), (Darboux 1896, chapter II), (Spivak 1999, vol. IV, chapter 7, Part F) and (Struik 1988, chapter 2).

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, 1998); a recent survey can be found in (Garcia and Sotomayor 2008).

The interest on the study of foliations with dense leaves goes back to Poincaré, Birkhoff, Denjoy, Peixoto, among others.

In

³ the asymptotic lines can be globally defined, an example is the Clifford torus, = ¹ (r) × ¹ (r) ⊂ ³, where ¹(r) = {(x, y) ∈ ²: x² + y² = r²} and r = /2. In all asymptotic lines are closed curves, in fact, Villarceau circles. (See Villarceau 1848) and illustration in Figure 1.

An asymptotic line γ is called recurrent if it is contained in the hyperbolic region and γ ⊆ L(γ), where L(γ) = α(γ) ∪ ω (γ) is the limit set of γ, and it is called dense if L(γ) = .

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.

2 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

Let v(u, v₀, є) be a solution of (1) with v(u, v₀, 0) = v₀ and u(u₀, v, є) solution of (1) with u(u₀,v,0) = u₀. Then the following variational equations holds:

PROOF. Differentiation with respect to є of (1) written as

taking into account that

leads to:

Analogous notation for b = b(u, v(u, v₀, є), є), c = c(u, v(u, v₀, є), є) and for the solution v (u, v₀, є).

Evaluation of equation (3) at є = 0 results in:

Differentiating twice the equation (1) and evaluating at є = 0 leads to:

Similar calculation gives the variational equations for and . This ends the proof.

□

3 DOUBLE RECURRENCE FOR ASYMPTOTIC LINES

Consider the Clifford torus = ¹ × ¹ ⊂ ³ parametrized by:

where is defined in the square Q = {(u, v): 0 < u < 2π, 0 < v < 2π}.

PROPOSITION 2. The asymptotic lines on the Clifford torus in the coordinates given by equation (5) are given by dud v = 0, that is, the asymptotic lines are the coordinate curves (Villarceau circles). See Figure 1.

PROOF. The coefficients of the first fundamental form I = Edu² + 2Fdudv + Gdv² and the second fundamental form II = edu² + 2 f dud v + gdv² 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 dud v = 0 and so they are the coordinate curves. Figure 1 is the image of the Clifford torus by a stereographic projection of ³ to ³.

□

THEOREM 1. There are embeddings α: ²→ ³such that all leaves of both asymptotic foliations,

_α,1and _α,2, are dense in . See Figure 2.

PROOF. Let N(u, v) = (α ∧ α_u∧ α_v) /|α ∧ α_u ∧ α_v|(u, v) be the unit normal vector to the Clifford torus.

We have that,

N(u, v) = (cos(–u + v), sin(–u + v), – cos(u, v), – sin(u + v)).

Let c(u, v) = h(u, v) N(u, v), h being a smooth 2π- double periodic function, and consider for є ≠ 0 small the one parameter family of embedded torus

Then the coefficients of the second fundamental form of with respect to

after multiplication by 1/(1 + є²h²)², are given by:

By Proposition 1 the variational equations of the implicit differential equation

with e(u, v, 0) = g(u ,v, 0) = 0, f(u, v, 0) = 1 and v(u, v₀, 0) = v₀ are given by:

In fact, differentiating equation (8) with respect to є it is obtained:

Making є = 0 leads to equation = 0.

Differentiation of equation (10) with respect to є and evaluation at є = 0 leads to

Therefore, the integration of the linear differential equations (9) leads to:

Taking h(u, v) = sin²(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).

So, from equation (11), it follows that

Therefore,

Consider the Poincaré map : {u = 0} → {u = 2π}, relative to the asymptotic foliation _α,1, defined by (v₀) = v(2π, v₀, є).

Therefore, = Id and it has the following expansion in є :

(v0) = v0 + ( 2π, v₀, 0) + O(є³) = v₀ – 12π є² + O(є³)

and so the rotation number of changes continuously and monotonically with є.

By the symmetry of the coefficients of the second fundamental form in the variables (u, v) and the fact that e(u, v, є) = g(u, v, є), see equation (12), it follows that the Poincaré map : {v = 0} → {v = 2π}, relative to the asymptotic foliation _α,2, defined by (u₀) = u(u₀, 2π, є) is conjugated to by an isometry.

Therefore we can take є ≠ 0 small such that the rotation numbers of , i = 1,2, are, modulo 2π, irrational. Therefore all orbits of πⁱ, i = 1,2, are dense in ¹. See (Katok and Hasselblatt 1995, chapter 12) or (Palis and Melo 1982, chapter 4). This ends the proof.

4 CONCLUDING COMMENTS

In this paper it was shown that there exist embeddings of the torus in

³ with both asymptotic 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. The same can be said for the technique of local bumpy perturbations of the Clifford Torus.

5 ACKNOWLEDGMENT

The authors are grateful to L.F. Mello and to the referees for their helpful comments. The authors are fellows of CNPq and done this work under the project CNPq no. 473747/2006-5.

Manuscript received on May 7, 2008; accepted for publication on September 22, 2008; contributed by JORGE SOTOMAYOR^*

AMS Classification: 53C12, 34D30, 53A05, 37C75.

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*

Member Academia Brasileira de Ciências

Publication Dates

• Publication in this collection
  19 Feb 2009
• Date of issue
  Mar 2009

History

• Received
  07 May 2008
• Accepted
  22 Sept 2008