Mean directionally curved lines on surfaces immersed in R4

Publ. Mat. 47 (2003), 415–440 MEAN DIRECTIONALLY CURVED LINES ON SURFACES IMMERSED IN R4 Luis Fernando Mello Abstract The notion of principal configuration of immersions of surfaces into R3, due to Sotomayor and Gutierrez [16] for lines of curvature and umbilics, is extended to that of mean directional configuration for immersed surfaces in R4. This configuration consists on the families of mean directionally curved lines, along which the second fundamental form points in the direction of the mean curvature vector, and their singularities, called here H-singularities. The concepts of H-singularities and periodic mean directionally curved lines are studied here in detail. Also the notion of principal structural stability of immersions of surfaces into R3is extended to that of mean directional structural stability, for the case of surfaces in R4. Sufficient conditions for immersions to be mean directional structurally stable are provided in terms of H-singularities, periodic mean directionally curved lines and the asymptotic behavior of all the other mean directionally curved lines. 1. Introduction Principal curvatures, principal direction fields, their integral foliations and umbilic singularities are classical topics of the theory of surfaces immersed in R3. Nevertheless the global behavior for a large class of these geometric objects has been understood only recently with the introduction, by Gutierrez and Sotomayor, of the notion of structural stability and genericity, originated from differential equations and dynamical systems. Their original works [16] and [10] are also presented in [11]. The global behavior of other geometric structures of the theory of surfaces immersed in R3, such as asymptotic lines [4] and lines of mean curvature [7], is still the subject of current works. 2000 Mathematics Subject Classification. 53C03, 58F14. Key words. Ellipse of curvature, minimal points, inflection points, normal curvature, structural stability. This work is supported in part by CNPq-Brazil, Grant 476886/2001–5. 416 L. F. Mello General aspects of the curvature theory for surfaces in R4are presented in the works of Forsyth [3], Wong [18], Little [12] and Asperti [1]. The ideas of Gutierrez and Sotomayor have also been applied to some aspects of this theory. Such as the description of generic singularities of asymptotic lines [5], the study of principal lines on minimal surfaces [8], and the analysis of generic singularities of lines of curvature, [9] and [15]. Recently the global behavior of lines of axial curvature on surfaces immersed in R4was studied by Garcia and Sotomayor [6]. The main feature of this paper is the study of the global generic structure of mean directionally curved lines on surfaces immersed in R4. Along these lines the second fundamental form points in the direction of the mean curvature vector. A review of properties of the first and second fundamental forms, the ellipse of curvature and related geometric objects is presented below. Afterwards, the main conclusions and the structure of this paper are formulated. In this paper immersions are assumed to be C∞. Nevertheless the results can be adapted for Crimmersions, r≥4. Let α:M→R4be an immersion of a smooth and oriented surface into R4, which is endowed with the euclidean inner product ·,· and oriented by a once for all fixed orientation. Denote respectively by TM and NM the tangent and the normal bundles of αand by TpMand NpMthe respective fibers, i.e., the tangent and the normal planes at p∈M. Let {N1,N 2}beaframe of vector fields orthonormal to α. Assume that (u, v)isapositive chart of Mand that {αu,α v,N 1,N 2}is a positive frame of R4.Insuch a chart (u, v) the first fundamental form of α,Iα,isgiven by I=Iα=dα, dα=Edu 2+2Fdudv+Gdv2, where E=αu,α u,F=αu,α vand G=αv,α v. The second fundamental form of α,IIα,isdefined in terms of the NM-valued quadratic form II =IIα=d2α, N1N1+d2α, N2N2=II1,αN1+II2,αN2, where IIi=IIi,α =eidu2+2fidu dv +gidv2, ei=αuu,N i,fi=αuv,N i, and gi=αvv,N i, for i=1,2. The following functions are associated to α(see [12]): 1. The mean curvature vector of α H=Hα=H1N1+H2N2, Mean Directionally Curved Lines 417 where Hi=Hi,α =Egi−2Ffi+Gei 2(EG −F2),for i=1,2. 2. The normal curvature of α kN=kN,α =E(f1g2−f2g1)−F(e1g2−e2g1)+G(e1f2−e2f1) 2(EG −F2). 3. The resultant ∆ofII1,α and II2,α ∆=∆ α=1 4(EG −F2) e12f1g10 e22f2g20 0e12f1g1 0e22f2g2  . 4. The normal curvature vector of αdefined by η:TM →NM, where η(p, v)=II(p,v) I(p,v). The image of the unitary circle S1of TpMby η(p): TpM→NpM describes an ellipse in NpMcalled ellipse of curvature of αat pand denoted by εα(p). This ellipse may degenerate into a line segment, a circle or a point. The center of the ellipse of curvature is the mean curvature vector Hand their area is given by π 2|kN(p)|. The map η(p) restricted to S1,being quadratic, is a double covering of the ellipse of curvature. Thus every point on the ellipse corresponds to two diametrically opposed points on the unitary tangent circle. The ellipse of curvature is invariant by rotations in the tangent and normal planes. Apoint p∈Mis called a minimal point of αif H(p)=0and it is called an inflection point of αif ∆(p)=kN(p)=0.Itfollows that p∈Mis an inflection point if and only if its ellipse of curvature is a radial line segment. Based in the above results we have: from any well-defined continuous choice of points on the ellipse of curvature, continuous tangent direction fields may be constructed on M.Ifthe construction fails for special points we say that they are singular points of the direction field. Consider the following construction. The line through the mean curvature vector H(p) meets the ellipse of curvature εα(p)attwo points. This construction induces two orthogonal directions on TpM. Making this construction for all p∈Mwe define two direction fields on M, called H-direction fields. The singularities of these fields, called H-singularities, are the points where either H=0(minimal 418 L. F. Mello points) or at which the ellipse of curvature becomes a radial line segment (inflection points). The set of H-singularities will be denoted by S(α). Amean directionally curved line is a regular curve ϕ:(a, b)→M which at each of its points is tangent to an H-direction and it contains any regular curve with this property which intersects it. The mean directionally curved lines and their singularities are assembled into the mean directional configuration. In this work, the notion of principal structural stability of immersions of surfaces into R3is extended for the mean directional configurations of the case R4.Sufficient conditions are provided to extend to the present setting the Theorem on Structural Stability for Principal Configurations due to Gutierrez and Sotomayor [11]. Two local cases, treated in detail here, are essential for this extension: H-singularities with their separatrix structure and closed (i.e., the cycles or periodic) mean directionally curved lines. This paper is organized as follows: In Section 2 we analyse the differential equation of mean directionally curved lines in an arbitrary chart. It is shown that this differential equation fits into the class of quadratic or binary differential equations. Section 3 is devoted to the analysis of H-singularities. For this purpose the differential equation of mean directionally curved lines is written in a Monge chart. The H-singularity condition is explicitly stated in terms of the coefficients of second order jet of the two functions which represent the immersion in a Monge chart. The condition of stability at H-singularity is expressed in an invariant form involving the third order jets. In Section 4 the derivative of first return map along a mean directionally curved cycle is established. It consists of an integral involving geometric functions along the cycle. In Section 5 the results presented in Sections 3 and 4 are put together to provide sufficient conditions for mean directional structural stability. In Section 6 we analyse the case where the surface Mis immersed in S3.Inthis case H-singularities only appear at inflection points. A correspondence between mean directionally curved lines on Mand lines of mean curvature on φ−1(M)isestablished, where φ:R3→S3is the stereographic projection. Furthermore it is shown that, for surfaces in S3, the quartic differential equation of lines of axial curvature factors into the quadratic differential equation of mean directionally curved lines and the quadratic differential equation of asymptotic lines. Mean Directionally Curved Lines 419 2. Differential equation for mean directionally curved lines The differential equation of mean directionally curved lines is given by η=µH,(2.1) where µ∈R. Eliminating µin (2.1) we have a quadratic differential equation of the form A(u, v)du2+2B(u, v)du dv +C(u, v)dv2=0,(2.2) where A=A(u, v)=(e1g2−e2g1)E+2(e2f1−e1f2)F,(2.3) B=B(u, v)=(f1g2−f2g1)E+(e2f1−e1f2)G,(2.4) C=C(u, v)=2(f1g2−f2g1)F+(e2g1−e1g2)G.(2.5) The H-singularities are determined by A=B=C=0in (2.2). But it is immediate that the equation EC =2FB−GA holds. We have established the following proposition. Proposition 2.1. Let α:M→R4be an immersion of a smooth and oriented surface into R4. With the above notations we have: 1. The differential equation of mean directionally curved lines is given by (2.2). 2. The H-singularities of αare given by A=B=0, where Aand B are defined in (2.3) and (2.4) respectively. Proposition 2.2. A(p)=B(p)=0if and only if pis either a minimal point or an inflection point of M. Proof: It is enough to prove the proposition for the isothermic coordinates where E=G=λ2and F=0.Inthis case equation (2.2) has the form A1(u, v)du2+2B1(u, v)du dv −A1(u, v)dv2=0, where A1=e1g2−e2g1, and B1=f1(e2+g2)−f2(e1+g1). 420 L. F. Mello Thus 0=A1f1+B1g1=(e1+g1)(f1g2−f2g1) and 0=A1f2+B1g2=(e2+g2)(f1g2−f2g1). If f1g2−f2g1=0then e1+g1=e2+g2=0and this implies that H(p)=0, i.e., pis a minimal point. If f1g2−f2g1=0then e1f2−e2f1= e1g2−e2g1=f1g2−f2g1=0and this implies that pis an inflection point. The reciprocal is immediate. From the Proposition 2.2, S(α)=S1(α)∪S2(α), where S1(α)isthe set of minimal points of Mand S2(α)isthe set of inflection points of M. The differential equation (2.2) is also obtained equivalently by Jac{Jac(II1,II 2),I}=0,(2.6) where Jac(·,·)= ∂(·,·) ∂(du, dv). Equation (2.6) suggests the definition of the quadratic form J=Jα= Jac(II1,II 2) =(e1f2−e2f1)du2+(e1g2−e2g1)du dv +(f1g2−f2g1)dv2. Denote by λ1(p) and λ2(p), with λ1(p)≤λ2(p), the extreme values of J as wranges on the unitary circle of TpM. These extreme values are the roots of λ2−2kNλ+∆=0. In fact, for w=a∂ ∂u +b∂ ∂v ,J(w,w) restricted to I(w,w)=1isstationary if and only if, for some (Lagrange multiplier) λ, ∂ ∂a(J(w,w))=λ∂ ∂a(I(w,w)) and ∂ ∂b(J(w,w))=λ∂ ∂b(I(w,w)).(2.7) Performing the differentiation and eliminating aand b,λwill satisfy the above equation. Thus the unitary vectors ±e1(p) and ±e2(p), at which the extreme values λ1(p) and λ2(p) are attained, are well defined. They are mutually orthogonal for poutside the set S(α), at which λ1(p)= λ2(p). Now, performing the differentiation of (2.7) and eliminating λ gives that, in the chart (u, v), the components aand bof ±e1(p) and ±e2(p) satisfies A(u, v)a2+2B(u, v)ab +C(u, v)b2=0, Mean Directionally Curved Lines 421 i.e., the differential equation of mean directionally curved lines (2.2). We define lH(α)=R(±e1) and LH(α)=R(±e2). If p∈ S(α) then (B2−AC)(p)>0. This implies the existence of two orthogonal solutions of the differential equation of mean directionally curved lines (2.2). Thus in a neighborhood of this point there exist two families of orthogonal curves. Under the orientability hypothesis it is possible to extend these lines to the whole M. Each family defines a foliation, denoted by FH(α) and fH(α) respectively, on the surface without the H-singularities. The foliation FH(α)(fH(α)) is associated to LH(α) (lH(α)). Each isolated H-singularity defines an isolated singularity of both foliations. The mean directional configuration of an immersion α:M→R4is the triple H(α)={S(α),F H(α),f H(α)}. It synthesizes the qualitative properties of the foliations FH(α) and fH(α) and represents the way their lines approach the H-singularity set. It is a natural analog of the principal configuration of an immersion α:M→R3. 3. Mean directional configurations near H-singularities Let p∈Mbe an H-singularity. We say that pis a transversal H-singularity if J(A, B)(p)=∂(A, B) ∂(u, v)(p)=0, where Aand Bare defined in (2.3) and (2.4) respectively. This condition means that the curves A=0and B=0,whose intersection defines the H-singularities, are regular and meet transversally at p.Itfollows that transversal H-singularities are isolated. Consider the surface Min a Monge chart, i.e., the surface Mis the graph of the map α(u, v)=(u, v, S(u, v),R(u, v)), where Sand Rare C∞functions defined on a neighborhood U⊂R2of (0,0) with the conditions S(0,0) = R(0,0) = Su(0,0) = Ru(0,0) = Sv(0,0) = Rv(0,0) = 0. For each point α(u, v)∈Mthe tangent plane Tα(u,v)Mis generated by {αu(u, v)=(1,0,S u(u, v),R u(u, v)),α v(u, v)=(0,1,S v(u, v),R v(u, v))} and the normal plane Nα(u,v)Mis generated by {¯ N1,¯ N2}, where 422 L. F. Mello ¯ N1=(−Su,−Sv,1,0) and ¯ N2=(−Ru(1 + Sv2)+SuSvRv, −Rv(1 + Su2)+SuSvRv,−SuRu−SvRv,1+Su2+Sv2). Therefore E=αu,α u,F=αu,α v,G=αv,α v, ei=αuu,N i,f i=αuv,N i,g i=αvv,N i, where Ni=¯ Ni ¯ Ni, for i=1,2. Write the Taylor’s expansion of the functions Sand Rnear (0,0) S(u, v)=s20 2u2+s11uv+s02 2v2+a 6u3+d 2u2v+b 2uv2+c 6v3+O(4),(3.1) R(u, v)=r20 2u2+r11uv+r02 2v2+¯a 6u3+¯ d 2u2v+¯ b 2uv2+¯c 6v3+O(4).(3.2) Thus the coefficients of the first and the second fundamental forms in a Monge chart are given by E=1+O(2),F=O(2),G=1+O(2), e1=s20 +au+dv+O(2),f 1=s11 +du+bv+O(2),g 1=s02 +bu+cv+O(2), e2=r20 +¯au+¯ dv+O(2),f 2=r11 +¯ du+¯ bv+O(2),g 2=r02 +¯ bu+¯cv+O(2). Mean Directionally Curved Lines 423 Define J=s02r20 −s20r02, K=s02¯a+r20b−s20¯ b−r02a, L=s02 ¯ d+r20c−s20¯c−r02d, M=s11(r20 +r02)−r11(s20 +s02), N=s11(¯a+¯ b)−r11(a+b)+(r20 +r02)d−(s20 +s02)¯ d, P=s11(¯c+¯ d)−r11(c+d)+(r20 +r02)b−(s20 +s02)¯ b. The differential equation of mean directionally curved lines (2.2) in a Monge chart has the form C(u, v)dv2+2B(u, v)du dv +A(u, v)du2=0,(3.3) where C(u, v)=J+Ku +Lv +Q1(u, v), B(u, v)=M+Nu+Pv+Q2(u, v), A(u, v)=−J−Ku −Lv +Q3(u, v), and Q1,Q2and Q3are of order O(2). From (3.3) the condition for (0,0) to be an H-singularity is that J=s02r20 −s20r02 =0 and M=s11(r20 +r02)−r11(s20 +s02)=0. Consider the tangent projective bundle PM ={TM −{0}/{v=rw, r=0}} of M. The natural projection is given by π:PM →M.We put p=dv/du and q=du/dv.ThusPM can be parametrized by charts (u, v;p) and (u, v;q). On PM consider the surface Wdefined by the differential equation of mean directionally curved lines. In coordinates (u, v;p) this surface is defined by T−1(0) where, from (3.3), (3.4) T(u, v;p)=(J+Ku +Lv +Q1)p2 +2(M+Nu+Pv+Q2)p−(J+Ku +Lv +Q3). If (0,0) is an H-singularity then (0,0; p)∈W, for all p.Furthermore the surface Wis smooth in a neighborhood of the p-axis if and only if (0,0) is a transversal H-singularity, according to [11] and [2]. 430 L. F. Mello Since H1(s)=0and (a(s))2+(b(s))2=0,for all s∈[0,L], we have that τg,2(s)=0,for all s∈[0,L]. Lemma 4.2. Let γbe amean directionally curved line and consider the chart (s, t)given in (4.1). Then the orthonormal frame {N1,N 2}of the normal bundle satisfies the following equations (N1)t(s, 0) = −τg,1(s)T1(s)−b(s)H1(s)T2(s)+a3 12(s)N2(s)(4.3) and (N2)t(s, 0) = −τg,2(s)T1(s)−a3 12(s)N1(s),(4.4) where a3 12(s)=a3 12(s, 0) = (N1)t(s, 0),N 2(s, 0)is the normal torsion of the frame {N1,N 2}associated to the mean directionally curved line orthogonal to γat the point γ(s). Proof: In a chart (s, t) the following equations hold (N1)t=g1F−f1G EG −F2αs+f1F−g1E EG −F2αt+a3 12N2 and (N2)t=g2F−f2G EG −F2αs+f2F−g2E EG −F2αt−a3 12N1. Using (4.2) the lemma is proved. Direct calculation shows that the following equations hold Et(s, 0) = −2kg,F t(s, 0) = Gt(s, 0) = 0, (e1)t(s, 0) = τ g,1−τg,2τn−kgH1(a+b), (f1)t(s, 0) = (bH1)+kgτg,1+τg,2a3 12, (g1)t(s, 0) = ¯ A, (e2)t(s, 0) = τg,1τn+τ g,2−aH1a3 12, (f2)t(s, 0) = kgτg,2−τg,1a3 12, (g2)t(s, 0) = ¯ B−bH1a3 12. Lemma 4.3. The function ¯ Bintroduced in (4.1) is given by ¯ B=2(H2)t+H1a3 12(a+b)−τg,1τn−τ g,2.(4.5) Mean Directionally Curved Lines 431 Proof: In the coordinates (s, t)wehave H2=Eg2−2Ff2+Ge2 2(EG −F2). Differentiating H2and using the above equations, the lemma is proved. Theorem 4.4. Let γbeamean directionally curved cycle of the foliation fH(α),parametrized by arc length sand of length L. Then the derivative of the first return map is given by π(0) = exp  1 2L 0  a(s)(H2)t(s)+H1(s)a3 12(s)−τg,1(s)τn(s) τg,2(s) ds . (4.6) Proof: The derivative of the first return map satisfies the following linear differential equation d ds dt dt0=−1 2B ∂A ∂t dt dt0 . Therefore π(0) = exp L 0 −At(s, 0) 2B(s, 0) ds,(4.7) where the functions Aand Bare given in (2.3) and (2.4). Using (4.2), (4.3) and (4.4) we have 2B(s, 0) = −2H1(s)τg,2(s)(a(s)+b(s)),(4.8) and At(s, 0) = H1(s)a(s)¯ B(s)−b(s)τg,1(s)τn(s)+τ g,2(s).(4.9) Substituting (4.5), (4.8) and (4.9) in (4.7) the theorem follows. Remark 4.5.The corresponding formula for the first derivative of the first return map when γis a mean directionally curved cycle of the foliation FH(α)isgiven by π(0) = exp  1 2L 0  b(s)(H2)t(s)+H1(s)a3 12(s)−τg,1(s)τn(s) τg,2(s) ds . (4.10) 432 L. F. Mello Proposition 4.6. Let α:M→R4be an immersion of a smooth and oriented surface into R4, and let γbeamean directionally curved cycle of the foliation fH(α),parametrized by arc length sand of length L. Consider a chart (s, t)as in the Lemma 4.1 and consider the deformation β(s, t)=β(, s, t)=α(s, t)+a(s)τg,2(s) 6t3δ(t)N2(s),(4.11) where δ=1in neighborhood of t=0, with small support and a≡ 0. Then γis a mean directionally curved cycle of β, for all ≥0small, and γis a hyperbolic mean directionally curved cycle for β,=0. Proof: From direct calculation with the deformation βit follows that γ is a mean directionally curved cycle for all β, and at t=0wehave 2(H2)t(s, 0) = τg,1τn+τ g,2−2H1a3 12 +¯ B+aτg,2. Therefore, assuming a≡ 0, it results that d d(ln π(0))=0 =1 2L 0 (a(s))2τg,2(s) 2τg,2(s)ds =1 4L 0 (a(s))2ds =0. 5. Mean directional structural stability Let I(M,R4)bethe space of the immersions of Minto R4, where M is a compact, smooth and oriented surface, endowed with the Whitney topology. Lemma 5.1. Let Mbeasurface which is the graph of the map α(u, v)= (u, v, S(u, v),R(u, v)),asinthe Section 3. Suppose that (0,0) is an H-singularity. We call βab the immersion β(α;u, v;a, b)=u, v, S(u, v)+a 2u2+b 2v2,R(u, v)+auv +b 2u2, with (a, b)∈V, where Vis a neighborhood of (0,0) and (u, v)∈U. Then there exists D⊂Uacompact disc on which a0(u, v)=∂(C, B) ∂(a, b)(u, v;0,0) =0, where Cand Bare obtained from the differential equation of mean directionally curved lines of βab. Proof: The differential equation of mean directionally curved lines of βab is given by C(u, v;a, b)(dv)2+2B(u, v;a, b)du dv +A(u, v;a, b)(du)2=0, Mean Directionally Curved Lines 433 where C=C(u, v;a, b)=2(f1g2−f2g1)F+(e2g1−e1g2)G(u, v;a, b) and B=B(u, v;a, b)=(f1g2−f2g1)E+(e2f1−e1f2)G(u, v;a, b). The form of A(u, v;a, b)isnot important here. Without loss of generality we have: 1. If the H-singularity is a minimal point then through an appropriate rotation in the normal plane it is possible to consider the frame {N1,N 2}and the principal axes of the ellipse of curvature as being parallels. This implies that e2=g2=f1=0and f2g1=0. 2. If the H-singularity is an inflection point then through an appropriate rotation in the normal plane it is possible to consider the ellipse of curvature on N1-direction. This implies that e2=f2=g2=0, g1=0and e1+g1=0. From extensive calculation we have a0(0,0) = ∂(C, B) ∂(a, b)(0,0; 0,0) = (e2+g1)(e1+g1+f2)=0. Therefore there exists D⊂Ua compact disc on which a0(u, v)=∂(C, B) ∂(a, b)(u, v;0,0) =0. This ends the proof. Lemma 5.2. Let Mbeasurface which is the graph of the map α(u, v)= (u, v, S(u, v),R(u, v)). Suppose that (0,0) is an H-singularity and let βab be the immersion β(α;u, v;a, b)=u, v, S(u, v)+a 2u2+b 2v2,R(u, v)+auv +b 2u2, with (a, b)∈V, where Vis a neighborhood of (0,0) and (u, v)∈U.Let D⊂Ube acompact disc on which a0(u, v)=0, as in the Lemma 5.1. Call Vab =Vab(D)⊂Vthe set of the pairs (a, b) for which all H-singularities of the immersion βab, with (u, v)∈D, satisfy the transversality condition. There is a small ρ>0 such that the intersection of Vab with the disc of radius ρhas full Lebesgue measure. 434 L. F. Mello Proof: The set Sab =S(βab)=(u, v;a, b)∈R2×R2/C(u, v;a, b)=0=B(u, v;a, b) is the locus of points where the immersion βab has H-singularities. Here Cand Bare the coefficients of the differential equation of mean directionally curved lines of βab. At transversal H-singularities Sab is a smooth surface, since at these points ∂(C, B) ∂(u, v)=0. Now we prove that if (a, b)isclose to (0,0), the surface Sab is regular even at non transversal H-singularities. From the Lemma 5.1, there exists D⊂Ua compact disc on which a0(u, v)=∂(C, B) ∂(a, b)(u, v;0,0) =0. This shows that there exists a neighborhood Vab ={|(a, b)|<ρ}such that if (u, v)∈Dand (a, b)∈Vab then ∂(C, B) ∂(a, b)(u, v;a, b)=0. This concludes the proof of the smoothness of Sab.Now the lemma is a consequence of Sard’s Theorem, by the identification of Vab with the regular values of the orthogonal projection of Sab to the ab-plane. Theorem 5.3. The set α∈I(M,R4)of immersions such that all H-singularities are of S-type is open and dense in I(M,R4). Proof: The conditions imposed to the immersion αnear an H-singularity of S-type depend on the derivatives of order three and are open, so it implies that this set is open. If the transversality condition holds for an H-singularity, a small perturbation on the parameters of the coordinate chart defines an H-singularity of S-type. Thus it is enough to consider H-singularities for which the transversality condition holds. The set of H-singularities of an immersion αis compact, therefore it can be covered by a finite number of Monge charts. Using the local Lemma 5.2, which can be globalized by a standard argument, the theorem is proved. Mean Directionally Curved Lines 435 An immersion αis said to be mean directional stable if it has a neighborhood V(α), such that for any β∈V(α) there exists a homeomorphism h:M→Mmapping S(α)ontoS(β), mapping FH(α)ontoFH(β) and mapping fH(α)ontofH(β). Consider the subset Σ ⊂I(M,R4)ofimmersions αdefined by the following conditions: 1. All H-singularities are S-type (Section 3). 2. All mean directionally curved cycles are hyperbolic (Section 4). 3. The limit set of every mean directionally curved line is contained in the set of H-singularities and mean directionally curved cycles. 4. There are no connections or self connections of H-singularity separatrices (Section 3). Theorem 5.4. The set Σis open in I(M,R4)and every α∈Σis mean directional stable. Proof: We take on the projective bundle PM,asinSection 3, the surface Wαdefined by the differential equation of mean directionally curved lines (2.2). In coordinates (u, v;p) this surface is defined by Wα=T−1 α(0), where Tα(u, v;p)=C(u, v)p2+2B(u, v)p+A(u, v). This surface is regular under the H-singularity hypothesis. The restriction of the natural projection π:PM →Mto Wαis a double covering outside the preimage of the set S(α). On PM we define the involution I(u, v;[du :dv])=(u, v;[dv :−du]) which amounts to a rotation of lines by an angle π/2. The surface Wαis invariant under I.OnWα −π−1 (S(α)) we define the vector field Xα, which in the coordinates (u, v;p) has the form Xα=((Tα)p,p(Tα)p,−((Tα)u+ p(Tα)v)). This vector field has an unique regular extension to π−1(S(α)). We consider the induced line field I∗Xα.Thusitisobtained a transversal pair {Xα,I ∗Xα}on Wα−π−1(S(α)). Therefore we have defined a net outside π−1(S(α)), with the following properties: this net is invariant under Iand by πprojects to the net (FH(α),f H(α)). With the above constructions this situation is connected with the case of the principal line fields and their canonical regions [11]. Thus the construction and continuation to a small neighborhood V(α)ofαof canonical regions follow also from the openness and unique continuation, for βnear α,ofsingularities and of cycles due to the hyperbolicity of these elements in the fields of the pair {Xα,I ∗Xα}. This leads of the openness of Σ and gives uniquely a correspondence between H-singularities, separatrices, cycles and their intersections for {Xα,I ∗Xα}and 436 L. F. Mello {Xβ,I ∗Xβ}. The extension of this correspondence to a topological equivalence H:Wα→Wβwhich, by projection, gives the topological equivalence h:M→Mbetween H(α) and H(β)iscarried out as in the case of nets of asymptotic lines on surfaces immersed in R3, according to [4]. 6. A special case Let α:M→S3be an immersion of a smooth and oriented surface into S3. Consider the natural inclusion i:S3→R4and the composition α=i◦αstill denoted by α. Assume that (u, v)isapositive chart of Mand that {αu,α v,N 1,N 2}is a positive frame of R4,{N1,N 2}being a frame of vector fields orthonormal to α, where N1(p)∈TpS3and N2(p)isthe inward normal to S3, for all p∈M.ThusN2≡−α. In such a chart (u, v) e2=E, f2=Fand g2=G, where E,Fand Gare the coefficients of the first fundamental form of α. It follows that II2=I.Now η=II I=II1 IN1+II2 IN2=II1 IN1+N2. This implies that the ellipse of curvature is degenerate as a line segment on N2=1,for all p∈M.Inclassic literature, this type of points are called semiumbilics and this result has been already obtained in [14]. We have H2=Eg2−2Ff2+Ge2 2(EG −F2)=1, for all p∈M.Itfollows that H(p)=0,for all p∈M. So, if pis an H-singularity of Mthen pis an inflection point of M.Inthis point the ellipse of curvature becomes a point. As an example consider the following construction. Let φ:R3→S3⊂ R4be the stereographic projection given by φ(x, y, z)= 1 1+w(x, y, z, w), where w=1 2(x2+y2+z2−1). We recall that φis conformal. Let α:M→R3be an immersion of a smooth and oriented surface M into R3. Assume that (u, v)isapositive chart of Mand that {αu,α v,N} is a positive frame of R3, where N=αu∧αv αu∧αv Mean Directionally Curved Lines 437 is the normal vector field to α. Let ¯α=φ◦αbe the stereographic projection of Min S3and let ¯α=i◦¯αbe the immersion of Minto R4, where {¯αu,¯αv,N 1,N 2}is a positive frame of R4,being N1=dφ(N) dφ(N) and N2the inward unitary normal to S3. From extensive calculation ¯ E=(1+w)−2E, ¯ F=(1+w)−2F, ¯ G=(1+w)−2G, ¯e1=(1+w)−2[(1 + w)e+Et], ¯ f1=(1+w)−2[(1 + w)f+Ft], ¯g1=(1+w)−2[(1 + w)g+Gt], ¯e2=¯ E, ¯ f2=¯ F, and ¯g2=¯ G, where the expressions without (with respectively) bar are associated to α (¯αrespectively), and t=α, Nis the support function of α. Lines of axial curvature on surfaces immersed in R4are lines along which the second fundamental form points in the direction of principal axes of the ellipse of curvature. The differential equation of lines of axial curvature is given by [6], Jacη−H2,I=0,(6.1) which is a quartic differential equation. Asymptotic lines on surfaces immersed in R4are lines along which the second fundamental form points in the direction of the tangent lines to the ellipse of curvature. The differential equation of asymptotic lines is given by Jac(II1,II 2)=0.(6.2) 438 L. F. Mello Remark 6.1.Through the above construction lines of principal curvature of αare carried over into asymptotic lines of ¯α, lines of mean curvature of αare carried over into mean directionally curved lines of ¯αand umbilic points of αare carried over into inflection points of ¯α. These results are also presented in [12] and [13]. Examples of immersions β∈Σ can be obtained from the Remark 6.1. Consider the set Ψ of immersions of surfaces into R3where every α∈Ψ is mean curvature structurally stable, according to [7]. Thus if α∈Ψ then ¯α∈Σ, where ¯α=i◦φ◦αis as above. Let α:M→R4be an immersion of a smooth and oriented surface into R4. The quartic differential equation (6.1) can be written as the product of two quadratic differential equations if the image of the surface Mby αis contained into R3, according to [6]. We have the following theorem. Theorem 6.2. Let α:M→S3be an immersion of a smooth and oriented surface into S3. Consider the natural inclusion i:S3→R4and the composition α=i◦αstill denoted by α. Then the quartic differential equation (6.1) canbewritten as Jac{Jac(II1,I),I}Jac(II1,I)=0,(6.3) where the first expression in (6.3) is the quadratic differential equation of mean directionally curved lines (2.6) and the second one is the quadratic differential equation of asymptotic lines (6.2). Proof: From the coefficients of the first and the second fundamental forms listed above write the product Jac{Jac(II1,I),I}Jac(II1,I) and compare the result with Jacη−H2,I. The theorem is proved. References [1] A. C. Asperti, Immersions of surfaces into 4-dimensional spaces with nonzero normal curvature, Ann. Mat. Pura Appl. (4) 125 (1980), 313–328. [2] J. W. Bruce and D. L. Fidal,Onbinary differential equations and umbilics, Proc. Roy. Soc. Edinburgh Sect. A 111(1–2) (1989), 147–168. [3] A. R. Forsyth,“Geometry of four dimensions”,vols. I and II, Cambridge Univ. Press, Cambridge, 1930. [4] R. Garcia, C. Gutierrez and J. Sotomayor, Structural stability of asymptotic lines on surfaces immersed in R3,Bull. Sci. Math. 123(8) (1999), 599–622. Mean Directionally Curved Lines 439 [5] R. Garcia, D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, Inflection points and topology of surfaces in 4-space, Trans. Amer. Math. Soc. 352(7) (2000), 3029–3043. [6] R. Garcia and J. Sotomayor, Lines of axial curvature on surfaces immersed in R4,Differential Geom. Appl. 12(3) (2000), 253–269. [7] R. Garcia and J. Sotomayor, Structurally stable configurations of lines of mean curvature and umbilic points on surfaces immersed in R3,Publ. Mat. 45(2) (2001), 431–466. [8] I. Guadalupe, C. Gutierrez, J. Sotomayor and R. Tribuzy, Principal lines on surfaces minimally immersed in constantly curved 4-spaces, in: “Dynamical systems and bifurcation theory” (Rio de Janeiro, 1985), Pitman Res. Notes Math. Ser. 160, Longman Sci. Tech., Harlow, 1987, pp. 91–120. [9] C. Gutierrez, I. Guadalupe, R. Tribuzy and V. Gu´ ı˜ nez, Lines of curvature on surfaces immersed in R4,Bol. Soc. Brasil. Mat. (N.S.) 28(2) (1997), 233–251. [10] C. Gutierrez and J. Sotomayor,Anapproximation theorem for immersions with stable configurations of lines of principal curvature, in: “Geometric dynamics” (Rio de Janeiro, 1981), Lecture Notes in Math. 1007, Springer, Berlin, 1983, pp. 332–368. [11] C. Gutierrez and J. Sotomayor, Lines of curvature and umbilical points on surfaces, 18th Brazilian Math. Colloquium, IMPA, Rio de Janeiro (1991); reprinted as: Structurally stable configurations of lines of curvature and umbilic points on surfaces, Monograf´ıas del IMCA, Lima (1998). [12] J. A. Little,Onsingularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl. (4) 83 (1969), 261–335. [13] D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, Osculating hyperplanes and asymptotic directions of codimension two submanifolds of Euclidean spaces, Geom. Dedicata 77(3) (1999), 305–315. [14] M. C. Romero-Fuster and F. S´ anchez-Bringas,Umbilicity of surfaces with orthogonal asymptotic lines in R4,Differential Geom. Appl. 16(3) (2002), 213–224. [15] F. S´ anchez-Bringas and A. I. Ram´ ırez-Galarza, Lines of curvature near umbilical points on surfaces immersed in R4,Ann. Global Anal. Geom. 13(2) (1995), 129–140.