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 GarciaI; Jorge SotomayorII
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 Correspondence to: Ronaldo Garcia E-mail: ragarcia@mat.ufg.br
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 Cr, 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):
Iα (p; v, w) = 〈Dα(p; v), Dα(p; w)〉,
IIα (p; v, w) = 〈-DNα(p; v), Dα(p; w)〉.
Here, Nα is the positive unit normal of the immersion a and 〈Nα, 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 Nα) 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 ωα,1, ωα,2 which locally split IIα as the product of ωα,1 and ωα,2.
The forms ωα,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 Cr-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 ∧ αu ∧ αv, 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 ∧ Cu ∧ Cv 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 ∧ αu∧ αv) /|α ∧ αu ∧ αv|(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* * Member Academia Brasileira de Ciências
AMS Classification: 53C12, 34D30, 53A05, 37C75.
- DO CARMO M. 1976. Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood-Cliffs.
- DARBOUX G. 1896. Leçons sur la Théorie des Surfaces, vol. II, Paris: Gauthier Villars.
- GARCIA R AND SOTOMAYOR J. 1997. Structural Stability of Parabolic Points and Periodic Asymptotic Lines, Mat Contemp 12: 83-102.
- GARCIA R AND SOTOMAYOR J. 2008. Lines of Curvature on Surfaces, Historical Comments and Recent Developments. Săo Paulo J Math Sci 2: 99-143.
- GARCIA R, GUTIERREZ C AND SOTOMAYOR J. 1999. Structural Stability of asymptotic lines on surfaces immersed in R3 B Sci Math 123: 599-622.
- GARCIA R, GUTIERREZ C AND SOTOMAYOR J. 1991. Lines of Curvature and Umbilic Points on Surfaces, 18th Brazilian Math Colloquium, IMPA, Rio de Janeiro, RJ, Brasil.
- GARCIA R, GUTIERREZ C AND SOTOMAYOR J. 1998. Structurally Configurations of Lines of Curvature and Umbilic Points on Surfaces. Instituto de Matemática y Ciencias Afines, IMCA, Lima, Universidad Nacional de Ingeniería, ISBN: 9972-753-01-8 84 p.
- KATOK A AND HASSELBLATT B. 1995. Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, England, UK.
- PALIS J AND DE MELO W. 1982. Geometric theory of dynamical systems. An introduction. Springer Verlag, New York, USA.
- SPIVAK M. 1999. A Comprehensive Introduction to Differential Geometry, Vol. III, chapter 2, p. 64 and Vol. IV, chapter 7, Part F, Berkeley, Publish or Perish.
- STRUIK D. 1988. Lectures on Classical Differential Geometry, chapter 2, Addison Wesley Pub Co, Reprinted by Dover Publications, Inc.
- VILLARCEAU Y. 1848. Note concernant un troisičme systčme de sections circulaires qu'admet le tore circulaire ordinaire. CR I-Math 27: 246.
Correspondence to:
Publication Dates
-
Publication in this collection
19 Feb 2009 -
Date of issue
Mar 2009
History
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Received
07 May 2008 -
Accepted
22 Sept 2008