A New Projective Invariant for Swallowtails and Godrons (Cusps of Gauss), and Global Theorems on the Flecnodal Curve 6 by Ricardo Uribe-Vargas∗ 0 Coll`ege deFrance, 3 rue d’Ulm, 75005 Paris. 0 2 [email protected] www.math.jussieu.fr/∼uribe/ n a J Abstract. We show some generic (robust) properties of smooth surfaces immersed in the 2 real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also 1 cusp of Gauss): an isolated parabolic point at which the (unique) asymptotic direction is tangenttotheparaboliccurve. Withthehelpofthesepropertiesandaprojectiveinvariant ] G thatweassociatetoeachgodronwepresentallpossiblelocalconfigurationsoftheflecnodal curve at a generic swallowtail in R3. We present some global results, for instance: In D a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of . transverse self-intersections (hence at least one self-intersection). h t Keywords: Geometryofsurfaces,Tangentialsingularities,swallowtail, paraboliccurve,flec- a m nodal curve,cusp of Gauss, godron, wavefront, Legendrian singularities. [ MSC2000: 14B05, 32S25, 58K35, 58K60, 53A20, 53A15, 53A05, 53D99, 70G45. 3 v 1 Introduction 4 8 4 A generic smooth surface in R3 has three (possibly empty) parts: an open 2 hyperbolic domain at which the Gaussian curvatureK is negative, an open el- 1 lipticdomainatwhichK ispositiveandaparabolic curveatwhichK vanishes. 4 0 A godron is a parabolic point at which the (unique) asymptotic direction is / h tangent to the parabolic curve. We present various robust geometric prop- t ertiesofgenericsurfaces,associatedtothegodrons. Forexample(Theorem3): a m : Any smooth curve of a surface of R3 tangent to the parabolic curve at a godron v g has at least 4-point contact with the tangent plane of the surface at g. i X The line formed by the inflection points of the asymptotic curves in the r a hyperbolic domain is called flecnodal curve. The next theorem is well known. Theorem 1. ([22, 14, 21, 17, 7]) At a godron of a generic smooth surface the flecnodal curve is (simply) tangent to the parabolic curve. Foranygenericsmoothsurfacewehavethefollowingglobalresult(Propo- sition 5 and Theorem 10): A closed parabolic curve bounding a hyperbolic disc has a positive evennumber of godrons, and the flecnodal curve lying in that disc has an odd number of transverse self-intersections (thus at least one self-intersection point). ∗Partially supported by EU Centre of Excellence, IMPAN-Banach Centre, ICA1-CT- 2000-70024. 1 The conodal curve of a surface S is the closure of the locus of points of contact of S with its bitangent planes (planes which are tangent to S at least at two distinct points). It is well known ([22, 14]) that: At a godron of a generic smooth surface the conodal curve is (simply) tangent to the parabolic curve. So the parabolic, flecnodal and conodal curves of a surface are mutually tangent atthegodrons. Ateach godron,thesethreetangent curvesdetermine aprojectiveinvariant ρ, asacross-ratio (seethecr-invariant below). We show all possible configurations of these curves at a godron, according to the value of ρ (Theorem 5). There are six generic configurations, see Fig. 2. Theinvariantρandthegeometricpropertiesofthegodronspresentedhere are useful for the study of the local affine (projective) differential properties of swallowtails. So, for example, we present all generic configurations of the flecnodal curve in the neighbourhood of a swallowtail point of a surface of R3 in general position (see Theorem 13 – Fig. 7 and Theorem 14 – Fig. 8). Our results are related to several mathematical theories as, for instance, implicit differential equations (Davidov [11]), contact geometry and Legen- drian singularities (Arnold [5, 6]), and more closely to differential geometry, singularities of projections and tangential singularities (Bruce, Giblin, Tari [9, 10], Goryunov [12], Landis [17], Platonova [21], Banchoff, Thom [8]). The paper is organised as follows. In section 2, we recall the classification of points of a generic smooth surface in terms of the order of contact of the surface with its tangent lines. In section 3, we give some definitions and present our results, proving some of them directly. Finally, in section 4, we give the proofs of the theorems. Acknowledgements. I would like to thank S. Janeczko, W. Domitrz and the Banach Centre for their hospitality and for the nice environment to do mathematics, to F. Aicardi andD.MeyerforusefulcommentsandtoE.GhysandD.Serreforthereferences[14,15,18]. 2 Projective properties of smooth surfaces Thepointsofagenericsmoothsurfaceinthereal3-space(projective, affineor Euclidean)areclassified intermsofthecontact of thesurfacewithits tangent lines. In this section, we recall this classification and some terminology. A generic smooth surface S is divided in three (possibly empty) parts: (E) An open domain of elliptic points: there is no real tangent line exceeding 2-point contact with S; (H) An open domain of hyperbolic points: there are two such lines, called asymptotic lines (their directions at the point of tangency are called asymp- totic directions); and (P) A smooth curve of parabolic points: a unique, but double, asymptotic line. The parabolic curve, divides S into the elliptic and hyperbolic domains. In the closure of the hyperbolic domain there is: (F) A smooth immersed flecnodal curve: it is formed by the points at which an asymptotic tangent line exceeds 3-point contact with S. 2 One may also encounter isolated points of the following four types: (g)Agodron isaparabolicpointatwhichthe(unique)asymptoticdirection is tangent to the parabolic curve; (hn) A hyperbonode is a point of the simplest self-intersection of the flecnodal curve; (b) A biflecnode is a point of the flec- nodal curve at which one asymptotic tangent exceeds 4-point contact with S (it is also called biinflection) ; (en) An ellipnode is a real point in the elliptic domain of the simplest self-intersection of the complex conjugate flecnodal curves associated to the complex conjugate asymptotic lines. In Fig. 1 the hyperbolic domain is represented in gray colour and the elliptic one in white. TheflecnodalcurvehasaleftbranchF (white)andarightbranchF (black). l r These branches will be defined in the next section. E en g P P H b F F b l r hn Figure 1: The 8 tangential singularities of a generic smooth surface. The term “godron” is due to R. Thom [13]. In other papers one can find the terms “special parabolic point” or “cusp of the Gauss map”. We keep Thom’s terminology since it is shorter. Here we will study the local projective differential properties of the godrons. The above 8 classes of tangential singularities, Theorem 1 and all the theorems presented in this paper are projectively invariant and are robust features of a smooth surface, that is, they are stable in the sense that under a sufficiently small perturbation (taking derivatives into account) they do not vanish but only deform slightly. Seven of these classes were known at the end of the 19th century in the context of the enumerative geometry of complex algebraic surfaces, with prominent works of Cayley, Zeuthen and Salmon, see [22]. For theseseven classes, thenormalformsofsurfaces atsuch pointsupto the 5-jet, under the group of projective transformations, were independently found by E.E. Landis ([17]) and O.A. Platonova ([21]). The ellipnodes were found by D. Panov ([20]) who called them special elliptic points. For surfaces in R3, these tangential singularities dependonly on the affine structure of R3 (because they depend only on the contact with lines), that is, they are independent of any Euclidean structure defined on R3 and of the GaussiancurvatureofthesurfacewhichcouldbeinducedbysuchaEuclidean structure. Another definition of godron (cusp of Gauss) is in terms of the contact with the tangent plane. In this setting, a useful tool for analysis is the so called‘heightfunction’(cf. [9,10]),whichcanbedefinedoncesomeEuclidean structure is fixed. Anyway, the singularities of the contact with the tangent 3 plane‘areexpressedgeometrically’, independentlyofanyEuclideanstructure, by the singularities of the dual surface: An ordinary godron corresponds to a swallowtail point of the dual surface, that is, to an A3 (Legendre) singularity (cf. [4, 6]). There are also double (unstable) godrons, corresponding to an A4 bifurcation (two swallowtails born or dying), that is, to an A4 (Legendre) singularity (see 3.5). § We will say that a godron is simple if it corresponds to a swallowtail point of the dual surface. All godrons of a surface in general position are simple. Both types of tangential singularities (contact with lines and contact with planes) were extensively used by Cayley, Zeuthen and Salmon, see [22]. Besides the smooth surfaces, we also consider surfaces admitting wave front singularities (section 3.7) and we study the behaviour of the flecnodal curve near the swallowtail points. 3 Statement of results Consider the pair of fields of asymptotic directions in the hyperbolic domain. An asymptotic curve is an integral curve of a field of asymptotic directions. Left and right asymptotic and flecnodal curves. Fix an orientation in the 3-space RP3 (or in R3). The two asymptotic curves passing through a point of the hyperbolic domain of a generic smooth surface can be distin- guished in a natural geometric way: Onetwists like a left screw and the other like a right screw. More precisely, a regularly parametrised smooth curve is said to be a left (right) curve if its first three derivatives at each point form a negative (resp. a positive) frame. Proposition 1. At a hyperbolic point of a surface one asymptotic curve is left and the other one is right. A proof is given (for generic surfaces) in Euclidean Remark below. The hyperbolic domain is therefore foliated by a family of left asymp- totic curves and by a family of right asymptotic curves. The corresponding asymptotictangentlinesarecalledrespectivelyleftandright asymptotic lines. Definition. The left (right) flecnodal curve F (resp. F ) of a surface S l r consistsofthepointsoftheflecnodalcurveofS whoseasymptoticline,having higher order of contact with S, is a left (resp. right) asymptotic line. Thefollowingstatement(complementtoTheorem1)isusedandimplicitly proved (almost explicitly) in [24, 26]. A proof is given in section 4, see Fig. 1: Theorem 2. A simple godron separates locally the flecnodal curve into its right and left branches. Definition. A flattening of a generic curve is a point at which the first three derivatives are linearly dependent. Equivalently, a flattening is a point at which the curve has at least 4-point contact with its osculating plane. The flattenings of a generic curve are isolated points separating the right and left intervals of that curve. 4 Euclidean Remark. If we fix an arbitrary Euclidean structure in the affine oriented space R3, then the lengths of the vectors and the angles between vectors are defined. Therefore, for such Euclidean structure, the torsion τ of a curve and the Gaussian curvature K of a surface are defined. In this case a point of a curve is right, left or flattening if the torsion at that point satisfies τ > 0, τ < 0 or τ = 0, respectively. The Gaussian curvature K on thehyperbolicdomainofasmoothsurfaceisnegative. TheBeltrami-Enepper Theorem states that the values of the torsion of the two asymptotic curves passing through a hyperbolic point with Gaussian curvature K are given by τ = √ K. This proves Proposition 1. ± − Definition. An inflection of a (regularly parametrised) smooth curve is a point at which the first two derivatives are linearly dependent. Equivalently, an inflection is a point at which the curve has at least 3-point contact with its tangent line. AgenericcurveintheaffinespaceR3 hasnoinflection. However, ageneric 1-parameter family of curves canhave isolated parameter values forwhichthe corresponding curve has one isolated inflection. Theorem 3. Let S be a smooth surface. All smooth curves of S which are tangent to the parabolic curve at a godron g have either a flattening or an inflection at g, and their osculating plane is the tangent plane of S at g. The proof of Theorem 3 is given in section 4. Corollary 1. If a point is a godron of a generic smooth surface, then it is a flattening of both the parabolic curve and the flecnodal curve. Remark. The converse is not true: A flattening of the parabolic curve or of the flecnodal curve is not necessarily a godron. 3.1 The cr-invariant and classification of godrons The conodal curve. Let S be a smooth surface. A bitangent plane of S is a plane which is tangent to S at least at two distinct points (which form a pair of conodal points). The conodal curve D of a surface S is the closure of the locus of points of contact of S with its bitangent planes. AtagodronofS,thecurveDissimplytangenttothecurvesP (parabolic) and F (flecnodal). This fact will be clear from our calculation of D for Platonova’s normal form of godrons. The projective invariant. At any simple godron g, there are three tangent smooth curves F, P and D, to which we will associate a projective invariant: Consider the Legendrian curves L , L , L and L (of the 3-manifold of F P D g contact elements of S, PT S) consisting of the contact elements of S tangent ∗ toF, P, D andtothepointg, respectively (thecontact elements of S tangent to a point are just the contact elements of S at that point, that is, L is the g fibre over g of the natural projection PT S S). These four Legendrian ∗ → curves are tangent to the same contact plane Π of PT S. The tangent di- ∗ rections of these curves determine four lines ℓ , ℓ , ℓ and ℓ , through the F P D g origin of Π. 5 Definition. The cr-invariant ρ(g) of a godron g is defined as the cross-ratio of the lines ℓ , ℓ , ℓ and ℓ of Π: F P D g ρ(g) = (ℓ ,ℓ ,ℓ ,ℓ ). F P D g Platonova’s normal form. According to Platonova’s Theorem [21], in the neighbourhood of a godron, a surface can be sent by projective transforma- tions to the normal form y2 2 4 1 z = x y+λx +ϕ(x,y) (for some λ = 0, ) (G1) 2 − 6 2 where ϕ is the sum of homogeneous polynomials in x and y of degree greater than 4 and (possibly) of flat functions. Theorem 4. Let g be a godron, with cr-invariant value ρ, of a generic smooth surface S. Put S (after projective transformations) in Platonova’s normal form (G1). Then the coefficient λ equals ρ/2. It turns out that among the 2-jets of the curves in S, tangent to P at a godron, there is a special 2-jet at which “something happens”. We introduce it in the following lemma. Tangential Map and Separating 2-jet. Let g be a godron of a generic smooth surface S. The tangential map of S, τ : S (RP3) , associates to S ∨ → each point of S its tangent plane at that point. The image S of τ is called ∨ S the dual surface of S. Write J2(g) forthesetof all2-jets ofcurvesofS tangent toP atg. Bythe image of a 2-jet γ in J2(g) under the tangential map τ we mean the image, S under τ , of any curve of S whose 2-jet is γ. By Theorem 3, all the 2-jets of S J2(g) (and also the 3-jets of curves on S tangent to P at g) are curves lying in the tangent plane of S at g. In suitable affine coordinates, the elements of J2(g) can be identified with the curves t (t,ct2,0), c R. 7→ ∈ Separating 2-jet Lemma. There exists a unique 2-jet σ in J2(g) (that we call separating 2-jet at g) satisfying the following properties: (a) The images, under τ , of all elements of J2(g) different from σ are cusps S of S sharing the same tangent line ℓ , at τ (g). ∨ ∨g S (b) The image of σ under τ is a singular curve of S whose tangent line at S ∨ τ (g) is different from ℓ . S ∨g (c) (separating property): The images under τ of any two elements of J2(g), S separated by σ, are cusps pointing in opposite directions. Remark. Once a godron with cr-invariant ρ of a smooth surface is sent (by projective transformations) to the normal form z = y2/2 x2y + ρx4/2 + − ϕ(x,y), the separating 2-jet is independent of ρ: It is given by the equation y = x2, in the (x,y)-plane. For almost all values of ρ the curves F, P and D are simply tangent one to the others. However, for isolated values of ρ two of these curves may have higher order of tangency and then some bifurcation occurs. We will look for the values of ρ at which ‘something happens’. 6 Canonicalcoefficients.Consideragodrong withcr-invariantρandsuppose that the surface was sent (by projective transformations) to the normal form z = y2/2 x2y +ρx4/2+ϕ(x,y). The plane curves F, P and D, which are − the projections of F, P and D to the (x,y)-plane along the z-axis, have the same 2-jet as F, P and D, respectively (since, according to Theorem 3, F, P and D have at least 4-point contact with the (x,y)-plane). These 2-jets correspond to three parabolas y = c x2, y = c x2 and y = c x2, whose F P D coefficients c , c and c we call the canonical coefficients of the curves F P D F, P and D, respectively. TheconfigurationofthecurvesF,P andDwithrespecttotheasymptotic lineandtheseparating2-jetatgisequivalenttotheconfigurationattheorigin of the curves F, P and D with respect to the parabolas y = 0 x2 = 0 and y = 1 x2 on the (x,y)-plane (see the above Remark). This con·figuration is · determined by the relative positions of the canonical coefficients c , c , c , F P D with respect to the numbers c = 1 and c = 0, in the real line: σ al Theorem 5. Given a simple godron g of a smooth surface, there are six possible configurations of the curves F, P and D with respect to the separating 2-jet and to the asymptotic line at g (they are represented in Fig. 2). The actual configuration at g depends on which of the following six open intervals the cr-invariant ρ(g) belongs to, respectively: ρ (1, ) 1 < c < c < c ; ∈ 2∞ ⇐⇒ D P F ρ ( ,1) 0 < c < c < c < 1; ∈ 13 2 ⇐⇒ P F D ρ ( , ) c < 0< c < c < 1; ∈ 2 13 ⇐⇒ P F D ρ (0, ) c < c < 0 < c < 1; ∈ 12 ⇐⇒ P F D ρ ( ,0) c < c < 0< c < 1; ∈ −2 1 ⇐⇒ P D F ρ ( , ) c < c < 0< 1 < c . ∈ −∞ −2 ⇐⇒ P D F 1<ρ 2 <ρ<1 1 <ρ< 2 3 2 3 0<ρ< 1 −1 <ρ<0 ρ<−1 2 2 2 Figure 2: Theconfigurations of thecurvesF (half-white half-black curves),P (boundary between white and gray domains), D (thick curves), the separating 2-jet (broken curves) and the asymptotic line (horizontal segments) at generic godrons. Besides the 5 exceptional values of ρ, given in Theorem 5, we will present separately ( 3.9.1) other important exceptional values of ρ. § 7 3.2 The index of a godron Definition. A godron is said to be positive or of index +1 (resp. negative or of index 1) if at the neighbouring parabolic points the half-asymptotic − lines, directed to the hyperbolic domain, point towards (resp. away from) the godron – Fig. 3 (some authors use the term hyperbolic (resp. elliptic)). (+) (−) Figure 3: A positive godron and a negative godron. The asymptotic double of the hyperbolic domain. A godron g can be positive or negative, depending on the index of the direction field, which is naturally associated to g, on the asymptotic double of S: The asymptotic A double of S is the surface in the manifold of contact elements of S, PT S, ∗ A consistingofthefieldofasymptoticdirections. Itdoublycoversthehyperbolic domain,anditsprojectiontoS hasafoldsingularityover theparaboliccurve. There is an asymptotic lifted field of directions on the surface , constructed A in the following way. At each point of the contact manifold PT S a contact ∗ plane is applied, in particular at each point of . Consider a point of the A smooth surface and assume that the tangent plane of at this point does A A not coincide with the contact plane. Then these two planes intersect along a straight line tangent to . The same holds at all nearby points in . This A A defines a smooth direction field on which vanishes only at the points where A those planes coincide: over the godrons. Ifg isapositivegodron,thentheindexofthisdirectionfieldatitssingular point equals +1, the point being a node or a focus; if g is negative, the index equals 1 and the point is a saddle. See Fig. 4. − (+) (+) (−) Figure 4: The asymptotic double of the hyperbolicdomain near a godron. Proposition 2. A godron g is positive (negative) if and only if the value of its cr-invariant ρ satisfies: ρ(g) > 1 (resp. ρ(g) < 1). We say that the elliptic domain of a smooth surface S is locally convex in the neighbourhood of a godron if, when projected to the tangent plane to S (from any point, exterior to this plane), the image of the elliptic domain is locally convex: Thetangentlineoftheparaboliccurvebeinglocally insidethe 8 image of the hyperbolic domain (the hyperbolic domain being locally convex if this line lies locally in inside the image of the elliptic domain). Corollary 2 (of Theorem 5). At a godron g with cr-invariant ρ the hyper- bolic (elliptic) domain is locally convex if and only if ρ > 2/3 (resp. ρ < 2/3). Proof. Thehyperbolic(elliptic)domainislocally convex atg ifandonlyifthe canonical coefficient c of the parabolic curve is positive (resp. negative). P Theorem 6. (a)Intheneighbourhood of anypositive godron thehyperbolic domain islocally convex. (b) There exist negative godrons for which the neighbouring hyperbolic domain is locally convex. (c) At the negative godrons for which the neighbouring hyperbolic domain is locally convex, the flecnodal curve lies locally between P and D (see Fig. 2). 2 Moreover, the cr-invariant satisfies: < ρ < 1. 3 Proof. The theorem follows from Proposition 2 and Corollary 2. Items (a) and (b) of Theorem 6 were discovered by F. Aicardi [1]. Corollary 3. All godrons of a cubic surface in RP3 are negative. Proof. By the definitions of asymptotic curve and of flecnodal curve, any straight line contained in a smooth surface is both an asymptotic curve and a connected component of the flecnodal curve of that surface. Let S be an algebraic surface of degree 3. At a point of the flecnodal curve, an asymptotic line has at least 4-point contact with S. Since S is a cubic surface, this line must lie completely in S. So the flecnodal curve of S consists of straight lines. At a godron g of S, the tangent line to the parabolic curve (that is, the flecnodal curve) lies in the hyperbolic domain. Thus the neighbouring elliptic domain is locally convex. Therefore, by the above theorem, g is negative. Factorisable polynomials. A set ℓ1,...,ℓn of real affine functions on { } the plane is said to be in general position if: (i) The lines ℓ = 0, ℓ = 0 are i j not parallel (i = j, i,j 1,...,n ), and (ii) For any i 1,...,n , the line 6 ∈ { } ∈ { } ℓ = 0 contains no critical point of the function ℓ . The product ℓ of i j=i j i n real affine functions in general position is calleQd a6 factorisable polynoQmial. In [19], A. Ortiz-Rodr´ıguez proved, among other things, that for any real factorisable polynomial of degree n, f = ℓ , the following holds: (i) The i lines ℓi are the only components of the flecQnodal curve of the graph of f, (ii) This graph has exactly n(n 2) godrons, and − Proposition 3. (Theorem 1 and Lemma 13 of [19]) All godrons of the graph of a real factorisable polynomial are negative. Proof. Theorem 6 provides an alternative and very simple proof of Proposi- tion 3: Since the flecnodal curve consists of straight lines, at each godron g the asymptotic tangent line and the asymptotic curve coincide with one of such straight lines. Thus, theelliptic domain is locally convex at g, andhence g is a negative godron (by Theorem 6). 9 3.3 Locating the left and right branches of the flecnodal curve Remark on the co-orientation of the elliptic domain. Each connected component of the elliptic domain is ‘naturally’ co-oriented: At each elliptic point the surface lies locally on one of the two half-spaces determined by its tangent plane at that point. This half-space, that we name positive half- space, determines a natural co-orientation on each connected component of the elliptic domain. By continuity, the natural co-orientation extends to the parabolic points (where the parabolic curve is smooth). At the parabolic points a positive half-space is therefore also defined. This observation has strong topological consequences. For example: Theorem 7. The elliptic domain of any smooth surface in the 3-space (Eu- clidean, affine or projective) can not contain a Mo¨bius strip. Proof. If a M¨obius strip M were contained in the elliptic domain E of a surface, then it would be contained in a connected component of E, since M is connected. Now, the theorem follows since each connected component of E has a natural co-orientation (and M is not co-orientable). In the neighbourhood of a godron g of a smooth surface S, we can distin- guish explicitly which branch of the flecnodal curve is the right branch and which is the left one. For this, we need only to know the index of g and the natural co-orientation of S (given by the positive half-space at g): Let g be a godron, with ρ = 1, of a smooth surface S. Take an affine 6 coordinate system x,y,z such that the (x,y)-plane is tangent to S at g, and the x-axis is tangent to the parabolic curve at g (thus also tangent to F at g). Directthepositivez-axis tothepositivehalf-spaceatg. Direct thepositivey- axis towards the neighbouring hyperbolic domain. Finally, direct the positive x-axis in such way that any basis (e ,e ,e ) of x,y,z form a positive frame x y z for the fixed orientation of R3 (or of RP3). So one can locally parametrise the flecnodal curve at g by projecting it to the x-axis. Theorem 8. Under the above parametrisation, the left and right branches of the flecnodal curve at g correspond locally to the negative and positive semi- axes of the x-axis, respectively, if and only if g is a positive godron. The opposite correspondence holds for a negative godron. In other words, if you stand on the tangent plane of S at g in the positive half-space and you are looking from the elliptic domain to thehyperbolicone, then you see the right (left) branch of the flecnodal curve on your right hand side if and only if g is a positive (resp. negative) godron. So the index of g determines and is determined by the side on which the right branch of F is located. Remark. Theorems 2 and 8 (which are local theorems) together with the naturalco-orientation oftheellipticdomain,arethekeyelements toprovethe global theorem (Theorem 10) of section 3.6. They imply that some (global) configurations of the flecnodal curve are forbidden. So, for example, there is no surface having a hyperbolic disc without hyperbonodes. 10