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1 Anais da Academia Brasileira de Ciências ISSN: Academia Brasileira de Ciências Brasil GARCIA, RONALDO; SOTOMAYOR, JORGE Tori embedded in S³ with dense asymptotic lines Anais da Academia Brasileira de Ciências, vol. 81, núm. 1, marzo, 009, pp Academia Brasileira de Ciências Rio de Janeiro, Brasil Available in: How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative

2 Anais da Academia Brasileira de Ciências (009) 81(1): (Annals of the Brazilian Academy of Sciences) ISSN Tori embedded in S 3 with dense asymptotic lines RONALDO GARCIA 1 and JORGE SOTOMAYOR 1 Instituto de Matemática e Estatística, Universidade Federal de Goiás Caixa Postal 131, Goiânia, GO, Brasil Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, Cidade Universitária, São Paulo, SP, Brasil Manuscript received on May 7, 008; accepted for publication on September, 008; contributed by JORGE SOTOMAYOR* ABSTRACT In this paper are given examples of tori T embedded in S 3 with all their asymptotic lines dense. Key words: asymptotic lines, recurrence, Clifford torus, variational equation. 1 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. The Fundamental Forms of α at a point p of M are the symmetric bilinear forms on T p M defined as follows (Spivak 1999): I α (p; v, w) = Dα(p; v), Dα(p; w), I I α (p; v, w) = DN α (p; v), Dα(p; w). 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, of the smooth one-forms ω α,1, ω α, which locally split I I α as the product of ω α,1 and ω α,. AMS Classification: 53C1, 34D30, 53A05, 37C75. *Member Academia Brasileira de Ciências Correspondence to: Ronaldo Garcia ragarcia@mat.ufg.br

3 14 RONALDO GARCIA and JORGE SOTOMAYOR 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, 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 on H α ; they are distinctly defined together with the ordering between them given by the subindexes {1, } which define their orientation ordering: 1 for the first asymptotic line field L α,1, for the second asymptotic line field L α,. The asymptotic foliations of α are the integral foliations A α,1 of L α,1 and A α, of L α, ; 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 I I = edu + f dudv + gdv = 0. In S 3, with the second fundamental form relative to the normal vector N = α α u α v, it follows that: e = det [ ] α, α u, α v, α uu, f = det [ ] α, α u, α v, α uv, g = det [ ] α, α u, α v, α vv. EG F EG F EG F 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, pg. 64). The study of asymptotic lines on surfaces M of R 3 and S 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 ). 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 008). The interest on the study of foliations with dense leaves goes back to Poincaré, Birkhoff, Denjoy, Peixoto, among others. In S 3 the asymptotic lines can be globally defined, an example is the Clifford torus, C = S 1 (r) S 1 (r) S 3, where S 1 (r) = {(x, y) R : x + y = r } and r = /. In C 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 ) = M. 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.

4 DENSE ASYMPTOTIC LINES ON TORI EMBEDDED IN S 3 15 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 a(u, v, ɛ)dv + b(u, v, ɛ)dudv + c(u, v, ɛ)du = 0, a(u, v, 0) = c(u, v, 0) = 0, b(u, v, 0) = 1. (1) Let v(u, v 0, ɛ) be a solution of (1) with v(u, v 0, 0) = v 0 u(u 0, v, 0) = u 0. Then the following variational equations holds: and u(u 0, v, ɛ) solution of (1) with c ɛ + v ɛu = 0, a ɛ + u ɛv = 0, c ɛɛ + c vɛ v ɛ b ɛ c ɛ + v uɛɛ = 0, a ɛɛ + a uɛ u ɛ b ɛ a ɛ + u vɛɛ = 0. () PROOF. Differentiation with respect to ɛ of (1) written as a(u, v, ɛ) taking into account that ( ) dv + b(u, v, ɛ) dv du du + c(u, v, ɛ) = 0, v(u, v 0, 0) = v 0, a v = a v, a ɛ = a ɛ, a ɛu = a uɛ = a ɛ u = a u ɛ, leads to: (a ɛ + a v v ɛ ) ( ) dv + a dv du du v ɛu + ( )dv b ɛ + b v v ɛ du + bv ɛu + c ɛ + c v v ɛ = 0. (3) Analogous notation for b = b(u, v(u, v 0, ɛ), ɛ), c = c(u, v(u, v 0, ɛ), ɛ) and for the solution v(u, v 0, ɛ). Evaluation of equation (3) at ɛ = 0 results in: c ɛ + v ɛu = 0. Differentiating twice the equation (1) and evaluating at ɛ = 0 leads to: c ɛɛ + c vɛ v ɛ + 4b ɛ v ɛu + bv uɛɛ = 0, c ɛɛ + c vɛ v ɛ b ɛ c ɛ + v uɛɛ = 0. Similar calculation gives the variational equations for u ɛ and u ɛɛ. This ends the proof. (4)

5 16 RONALDO GARCIA and JORGE SOTOMAYOR 3 DOUBLE RECURRENCE FOR ASYMPTOTIC LINES ( ) ( ) Consider the Clifford torus C = S 1 1 S 1 1 S 3 parametrized by: ( ) C(u, v) = cos( u + v), sin( u + v), cos(u + v), sin(u + v), (5) where C is defined in the square Q = { (u, v) : 0 u π, 0 v π }. PROPOSITION. 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 Figure 1. Fig. 1 Torus and Villarceau circles. PROOF. The coefficients of the first fundamental form I = Edu + Fdudv + Gdv and the second fundamental form I I = edu + f dudv+gdv of C with respect to the normal vector field N = C C u C v are given by: E(u, v) =1, e(u, v) = 0, F(u, v) =0, f (u, v) = 1, G(u, v) =1, g(u, v) = 0. Therefore the asymptotic lines are defined by dudv = 0 and so they are the coordinate curves. Figure 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 S 3 such that all leaves of both asymptotic foliations, A α,1 and A α,, are dense in T. See Figure. 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).

6 DENSE ASYMPTOTIC LINES ON TORI EMBEDDED IN S 3 17 Fig. Stereographic projection of a deformation of a Clifford torus with ɛ = /3. Let c(u, v) = h(u, v)n(u, v), h being a smooth π double periodic function, and consider for ɛ = 0 small the one parameter family of embedded torus α ɛ (u, v) = Then the coefficients of the second fundamental form of α ɛ with respect to after multiplication by 1/(1 + ɛ h ), are given by: C(u, v) + ɛc(u, v) C(u, v) + ɛc(u, v). (6) N ɛ = α ɛ (α ɛ ) u (α ɛ ) v / α ɛ (α ɛ ) u (α ɛ ) v, e = ɛh uu + ɛ h u h v + ɛ 3( hh u h h uu ), f = 1 + ɛh uv + ɛ ( h u + h v h) + ɛ 3( hh u h v h h uv ) + ɛ 4 h 4, (7) g = ɛh vv + ɛ h u h v + ɛ 3( hh v h h vv ). By Proposition 1 the variational equations of the implicit differential equation e(u, v, ɛ) + f (u, v, ɛ) dv ( ) dv + g(u, v, ɛ) = 0, (8) du du with e(u, v, 0) = g(u, v, 0) = 0, f (u, v, 0) = 1 and v(u, v 0, 0) = v 0 are given by: e ɛ + v ɛu = 0, e ɛɛ + e vɛ v ɛ f ɛ e ɛ + v uɛɛ = 0. (9) In fact, differentiating equation (8) with respect to ɛ it is obtained: (e v (u, v, ɛ)v ɛ + e ɛ (u, v, ɛ)) + ( f v (u, v, ɛ)v ɛ + f ɛ (u, v, ɛ))v u + f v uɛ (u, v, ɛ) + (g v (u, v, ɛ)v ɛ + g ɛ (u, v, ɛ))(v u ) + g(u, v, ɛ)v u v uɛ = 0. (10) Making ɛ = 0 leads to equation e ɛ + v ɛu = 0. Differentiation of equation (10) with respect to ɛ and evaluation at ɛ = 0 leads to e ɛɛ + e vɛ v ɛ f ɛ e ɛ + v uɛɛ = 0.

7 18 RONALDO GARCIA and JORGE SOTOMAYOR Therefore, the integration of the linear differential equations (9) leads to: v ɛ (u) = 1 u 0 h uu du, v uɛɛ = 1 h uuv u Taking h(u, v) = sin (v u), it results from equation (7) that: 0 h uu du + h uu h uv h u h v. (11) e(u, v, ɛ) = e(v, u, ɛ) = g(u, v, ɛ), f (u, v, ɛ) = f (v, u, ɛ). (1) In fact, from the definition of h it follows that: h(u, v) = h(v, u), h uu = h vv = 8 cos(4v 4u), h u = h v = sin(4v 4u), h uv = 8 cos(4v 4u). So, a careful calculation shows that equation (1) follows from equation (7). So, from equation (11), it follows that Therefore, v ɛ (u, v 0, 0) = sin(4v 0 ) sin(4v 0 4u), v ɛɛ (u, v 0, 0) = 1u 4 sin(4u) 4 sin(8v 0 4u), 5 sin(8v 0) + 13 sin(8v 0 8u). v ɛ (π, v 0, 0) v ɛ (0, v 0, 0) = 0, v ɛɛ (π, v 0, 0) v ɛɛ (0, v 0, 0) = 4π. Consider the Poincaré map πɛ 1 : {u = 0} {u = π}, relative to the asymptotic foliation A α,1, defined by πɛ 1(v 0) = v(π, v 0, ɛ). Therefore, π0 1 = I d and it has the following expansion in ɛ: πɛ 1 (v 0) = v 0 + ɛ v ɛɛ(π, v 0, 0) + O(ɛ 3 ) = v 0 1πɛ + O(ɛ 3 ) and so the rotation number of πɛ 1 changes continuously and monotonically with ɛ. By the symmetry of the coefficients of the second fundamental form in the variables (u, v) and the fact that e(u, v, ɛ) = g(u, v, ɛ), see equation (1), it follows that the Poincaré map πɛ : {v = 0} {v = π}, relative to the asymptotic foliation A α,, defined by πɛ (u 0) = u(u 0, π, ɛ) is conjugated to πɛ 1 by an isometry. Therefore we can take ɛ = 0 small such that the rotation numbers of πɛ i, i = 1,, are, modulo π, irrational. Therefore all orbits of π i, i = 1,, are dense in S 1. See (Katok and Hasselblatt 1995, chapter 1) or (Palis and Melo 198, chapter 4). This ends the proof. 4 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. 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.

8 DENSE ASYMPTOTIC LINES ON TORI EMBEDDED IN S 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 / RESUMO Neste artigo são dados exemplos de toros T mergulhados em S 3 com todas as suas linhas assintóticas densas. Palavras-chave: linhas assintóticas, recorrências, toro de Clifford, equação variacional. REFERENCES DO CARMO M Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood-Cliffs. DARBOUX G Leçons sur la Théorie des Surfaces, vol. II, Paris: Gauthier Villars. GARCIA R AND SOTOMAYOR J Structural Stability of Parabolic Points and Periodic Asymptotic Lines, Mat Contemp 1: GARCIA R AND SOTOMAYOR J Lines of Curvature on Surfaces, Historical Comments and Recent Developments. São Paulo J Math Sci : GARCIA R, GUTIERREZ C AND SOTOMAYOR J Structural Stability of asymptotic lines on surfaces immersed in R 3. B Sci Math 13: GUTIEREZ C AND SOTOMAYOR J Lines of Curvature and Umbilic Points on Surfaces, 18 th Brazilian Math Colloquium, IMPA, Rio de Janeiro, RJ, Brasil. GUTIERREZ C AND SOTOMAYOR J Structurally Configurations of Lines of Curvature and Umbilic Points on Surfaces. Instituto de Matemática y Ciencias Afines, IMCA, Lima, Universidad Nacional de Ingeniería, ISBN: p. KATOK A AND HASSELBLATT B Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, England, UK. PALIS J AND DE MELO W Geometric theory of dynamical systems. An introduction. Springer Verlag, New York, USA. SPIVAK M A Comprehensive Introduction to Differential Geometry, Vol. III, chapter, p. 64 and Vol. IV, chapter 7, Part F, Berkeley, Publish or Perish. STRUIK D Lectures on Classical Differential Geometry, chapter, Addison Wesley Pub Co, Reprinted by Dover Publications, Inc. VILLARCEAU Y Note concernant un troisième système de sections circulaires qu admet le tore circulaire ordinaire. CR I-Math 7: 46.