LINE FIELDS ON SURFACES IMMERSED IN IR 1

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sible that the lines of axial curvature have transversal crossings. General .... appropriate rotation of the normal plane it is possible to take N1(p) parallel to εα(p).
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LINE FIELDS ON SURFACES IMMERSED IN IR4

L. F. MELLO Instituto de Ciˆencias, Universidade Federal de Itajub´ a, CEP 37.500-903, Itajub´ a, MG, Brazil E-mail: [email protected]

In this paper we show that a necessary and sufficient condition for the differential equation of lines of axial curvature factors into differential equations of mean directionally curved lines and asymptotic lines is the vanishing of the normal curvature.

1. Introduction There exist three line fields naturally defined on a surface M in IR 4 : the lines of axial curvature, the mean directionally curved lines and the asymptotic lines. The lines of axial curvature are globally defined on surfaces in IR 4 and their singularities are the axiumbilic points where the ellipse of curvature becomes either a circle or a point. The axiumbilic points and the lines of axial curvature are assembled into two axial configurations. The first (second) one is defined by the axiumbilics and the field of orthogonal lines on which the surface is curved along the large (small) axis of the ellipse of curvature. A line of axial curvature is not necessarily a simple regular curve; it can be immersed with transversal crossings. The differential equation of lines of axial curvature is a quartic differential equation 1 . The mean directionally curved lines are globally defined on surfaces in IR4 and their singularities are either the inflection points, where the ellipse of curvature is a radial line segment, or the minimal points, where the mean curvature vector vanishes. The differential equation of mean directionally curved lines is a quadratic differential equation 2 . The asymptotic lines do not need to be globally defined on the surfaces and in general are not orthogonal. A necessary and sufficient condition for existence of the globally defined asymptotic lines on a surface M in IR 4 is the local convexity of M 3 . The differential equation of asymptotic lines is 951

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also a quadratic differential equation and their singularities are the inflection points 4 . In this paper we prove the following theorem: Let α : M → IR 4 be an immersion of a compact smooth oriented surface with isolated inflection points. The quartic differential equation of lines of axial curvature can be written as the product of the quadratic differential equations of mean directionally curved lines and asymptotic lines if and only if the normal curvature of α vanishes at every point. Therefore if the normal curvature vanishes at every point it is not possible that the lines of axial curvature have transversal crossings. General aspects of the curvature theory for surfaces immersed in IR 4 are presented in the works of Little 5 and Wong 6 . 2. Line fields on surfaces in IR4 In this paper immersions are assumed to be C ∞ . Let α : M → IR4 be an immersion of a compact smooth oriented surface into IR 4 , which is endowed with the Euclidean inner product h·, ·i and is oriented. Denote respectively by T M and N M the tangent and the normal bundles of α and by Tp M and Np M the tangent and the normal planes at p ∈ M . Let {N1 , N2 } be a frame of vector fields orthonormal to α. Assume that (u, v) is a positive chart of M and {αu , αv , N1 , N2 } is a positive frame of IR4 . In such a chart (u, v) the first fundamental form of α, Iα , is given by I = Iα = hdα, dαi = Edu2 + 2F dudv + Gdv 2 , where E = hαu , αu i, F = hαu , αv i and G = hαv , αv i. The second fundamental form of α, IIα , is defined in terms of the NM-valued quadratic form II = IIα = hd2 α, N1 iN1 + hd2 α, N2 iN2 = II1,α N1 + II2,α N2 , where IIi = IIi,α = ei du2 + 2fi dudv + gi dv 2 , ei = hαuu , Ni i, fi = hαuv , Ni i and gi = hαvv , Ni i, for i = 1, 2. The following functions are associated 5 to α: (1) The mean curvature vector of α H = Hα = H1 N1 + H2 N2 , where Hi = Hi,α =

Egi − 2F fi + Gei ; 2(EG − F 2 )

(2) The normal curvature of α kN = kN,α =

E(f1 g2 − f2 g1 ) − F (e1 g2 − e2 g1 ) + G(e1 f2 − e2 f1 ) ; 2(EG − F 2 )

(3) The resultant ∆ of II1,α and II2,α ∆ = ∆α =

4(f1 g2 − f2 g1 )(e1 f2 − e2 f1 ) − (e1 g2 − e2 g1 )2 ; 4(EG − F 2 )

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(4) The normal curvature vector of α defined by η(p, v) =

II(p,v) I(p,v) .

The image of the unitary tangent circle S 1 by η(p) : Tp M → Np M describes an ellipse in Np M called ellipse of curvature of α at p and 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 H and their area is given by π2 |kN (p)|. The map η(p) restricted to S 1 , 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 both the tangent and normal planes. A point p ∈ M is called a minimal point of α if H(p) = 0 and it is called an inflection point of α if ∆(p) = kN (p) = 0. It follows that p ∈ M is an inflection point if and only if εα (p) is a radial line segment 5 . From any well-defined continuous choice of points on the ellipse of curvature continuous tangent direction fields may be constructed on M . Also, from any well-defined continuous choice of diametrically opposed points on the ellipse of curvature continuous tangent orthogonal cross fields may be constructed on M . If the construction fails for special points of M we say that they are singular points of the fields. Lines of axial curvature. The four vertices of εα (p) determine eight points on the unitary tangent circle, which define two crosses in the tangent plane. Thus, we have two cross fields on M , called axial curvature cross fields. This construction fails at the axiumbilic points where the ellipse of curvature becomes either a circle or a point. The integral curves of the axial curvature cross fields are the lines of axial curvature. The differential equation of lines of axial curvature is a quartic differential equation 1   2 Jac kη − Hk , I = 0, (2.1) where Jac(·, ·) =

∂(·,·) ∂(du,dv) ,

which can be written as

A0 du4 + A1 du3 dv + A2 du2 dv 2 + A3 dudv 3 + A4 dv 4 = 0, A0 = a0 E 3 , A1 = a1 E 3 , A2 = −6a0 GE 2 + 3a1 F E 2 , A3 = −8a0 EF G+a1 E(4F 2 −EG), A4 = a0 G(EG−4F 2 )+a1 F (2F 2 −EG),

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  2 2 2 2 2 2 2 3 a0 = 4 F (EG−2F )(e1 +e2 )−E(EG−4F )a2 −E F (a3 +2(f1 +f2 ))+E a4 ,   2 2 2 3 2 2 2 2 2 a1 = 4 G(EG−4F )(e1 +e2 )+8EF Ga2 +E (g1 +g2 )−2E G(a3 +2(f1 +f2 )) , a2 = e 1 f1 + e 2 f2 , a3 = e 1 g1 + e 2 g2 , a4 = f 1 g1 + f 2 g2 . Generically there is no good way to distinguish one end of the large (or small) axis of εα (p) and therefore pick out a direction of the cross field. Thus a line of axial curvature is not necessarily a simple regular curve; it can be immersed with transversal crossings. Mean directionally curved lines. The line through the mean curvature vector H(p) meets εα (p) at two diametrically opposed points. This construction induces two orthogonal directions on Tp M . Therefore, we have two orthogonal direction fields on M , called H-direction fields. The singularities of these fields, called here H-singularities, are the points where either H = 0 (minimal points) or at which the ellipse of curvature becomes a radial line segment (inflection points). The integral curves of the H-direction fields are the mean directionally curved lines. The differential equation of mean directionally curved lines is 2 Jac{Jac(II1 , II2 ), I} = 0,

(2.2)

which can be written as B1 (u, v)du2 + 2B2 (u, v)dudv + B3 (u, v)dv 2 = 0, where B1 = (e1 g2 − e2 g1 )E + 2(e2 f1 − e1 f2 )F, B2 = (f1 g2 − f2 g1 )E + (e2 f1 − e1 f2 )G and B3 = 2(f1 g2 − f2 g1 )F + (e2 g1 − e1 g2 )G. It is possible to separate the mean directionally curved lines into two line fields 2 . Asymptotic lines. Suppose that the origin of Np M lies outside εα (p), for all p ∈ M . The two points on εα (p) where the lines through the normal curvature vectors are tangent to εα (p) induce a pair of directions on Tp M which in general are not orthogonal. Thus we have two direction fields on M , called asymptotic direction fields. The singularities of these fields are the points where the ellipse of curvature becomes a radial line segment, i.e., the inflection points. The integral curves of the asymptotic direction fields are the asymptotic lines. The differential equation of asymptotic lines is 2 Jac(II1 , II2 ) = 0,

(2.3)

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which can be written as T1 (u, v)du2 +T2 (u, v)dudv +T3 (u, v)dv 2 = 0, where T1 = e1 f2 − e2 f1 , T2 = e1 g2 − e2 g1 and T3 = f1 g2 − f2 g1 . It is possible to separate the asymptotic lines into two line fields. We have established 2 the following theorem: Let α : M → S 3 (r) be an immersion of a compact smooth oriented surface into a 3-dimensional sphere of radius r > 0. Consider the natural inclusion i : S 3 (r) → IR4 , and let the composition i◦α also be denoted by α. Then, the quartic differential equation of lines of axial curvature is the product of the quadratic differential equations of its mean directionally curved lines and of its asymptotic lines. In the above construction asymptotic lines are orthogonal and the normal curvature of α vanishes at every point. This is a particular case of the following lemma, which was also obtained in Ref. 7 using a different approach. Lemma 2.1. Let α : M → IR4 be an immersion of a compact smooth oriented surface with isolated inflection points. The immersion α has orthogonal asymptotic lines if and only if the normal curvature of α vanishes at every point. Proof. The asymptotic lines are orthogonal if and only if T1 ≡ −T3 for some chart, where T1 and T3 are the coefficients of the differential equation of asymptotic lines (2.3). But T1 ≡ −T3 is equivalent to f2 (e1 − g1 ) + f1 (g2 − e2 ) ≡ 0 which is equivalent to kN ≡ 0 in isothermic coordinates, where the coefficients of the first fundamental form are E = G 6= 0 and F = 0. Theorem 2.1. Let α : M → IR4 be an immersion of a compact smooth oriented surface with isolated inflection points. The quartic differential equation of lines of axial curvature (2.1) can be written as Jac{Jac(II1 , II2 ), I} · Jac(II1 , II2 ) = 0,

(2.4)

if and only if the normal curvature of α vanishes at every point. Proof. Suppose kN ≡ 0. In this case asymptotic and mean directionally curved lines are globally defined on M . The ellipse of curvature εα (p) is a line segment for all p ∈ M , except at the inflection points. Through an appropriate rotation of the normal plane it is possible to take N1 (p) parallel to εα (p). This implies that e2 = g2 6= 0 and f2 = 0. Thus the differential

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equations of mean directionally curved lines, asymptotic lines and lines of axial curvature can be written as   Ee2 (e1 − g1 )du2 + 4f1 dudv − (e1 − g1 )dv 2 = 0, (2.5)   e2 −f1 du2 + (e1 − g1 )dudv + f1 dv 2 = 0 4E

3



f 1 b0



(2.6)

     2 2 2 2 du −6du dv +dv + b0 −4f1 du −dv dudv = 0, (2.7) 4

2

2

4

b0 = g1 − e1 , in isothermic coordinates, respectively. Thus, a straightforward calculation yields that the differential equation (2.7) is the product of the differential equations (2.5) and (2.6). Conversely, if the differential equation of the lines of axial curvature is the product of the differential equations of mean directionally curved lines and asymptotic lines then, in particular, the asymptotic lines are orthogonal and, by lemma 2.1, we have that the normal curvature of α vanishes at every point. The theorem is thus proved.

Acknowledgments This work is supported in part by CNPq-Brazil, Grant 476886/2001-5. References 1. R. Garcia and J. Sotomayor, Differential Geom. Appl. 12, 253 (2000). 2. L. F. Mello, Pub. Math. 47, 415 (2003). 3. D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, Geom. Dedicata 54, 323 (1995). 4. R. Garcia, D. K. H. Mochida, M. C. Romero-Fuster and M. A. S. Ruas, Trans. Amer. Math. Soc. 352, 3029 (2000). 5. J. A. Little, Ann. Mat. Pura Appl. 83, 261 (1969). 6. W. C. Wong, Comment. Math. Helv. 26, 152 (1952). 7. M. C. Romero-Fuster and F. S´ anchez-Bringas, Differential Geom. Appl. 16, 213 (2002).