Title Singularities of ruled surfaces in [open face R]3. Author(s)IZUMIYA, SHYUICHI; TAKEUCHI, NOBUKO

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1 Title Singularities of ruled surfaces in [open face R]3 Author(s)IZUMIYA, SHYUICHI; TAKEUCHI, NOBUKO CitationMathematical Proceedings of the Cambridge Philosophi Issue Date 2 Doc URL Rights Copyright 2 Cambridge University Press Type article File Information MPCPS3.pdf Instructions for use Hokkaido University Collection of Scholarly and Aca

2 Mathematical Proceedings of the Cambridge Philosophical Society Vol. 3 January 2 Part Math. Proc. Camb. Phil. Soc. (2), 3, Printed in the United Kingdom c 2 Cambridge Philosophical Society Singularities of ruled surfaces in R 3 By SHYUICHI IZUMIYA Department of Mathematics, Hokkaido University, Sapporo 6-8, Japan. izumiya@math.sci.hokudai.ac.jp and NOBUKO TAKEUCHI Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 84-85, Japan. nobuko@u-gakugei.ac.jp (Received 2 May 999; revised September 999) Abstract We study singularities of ruled surfaces in R 3. The main result asserts that only cross-caps appear as singularities for generic ruled surfaces.. Introduction The study of ruled surfaces in R 3 is a classical subject in differential geometry. It has again been studied in some areas (i.e. Projective differential geometry [6], Computer-aided design [7, 8], etc.). Generally, ruled surfaces have singularities. Recently there have appeared several articles concerning singularities of developable surfaces in R 3 (cf. [3, 8 2, 4, 5, 7]). Developable surfaces are ruled surfaces which have vanishing Gauss curvature on the regular part. Another characterization is that developable surfaces are envelopes of one-parameter families of planes in R 3, so that they have singularities of discriminants of such families. In these articles classifications of singularities of developable surfaces are given. Briefly speaking, the cuspidal edge, the cuspidal cross-cap or the swallowtail appear as singularities of developable surfaces in general (cf. Fig. ). On the other hand, the Gaussian curvature of the regular part of a ruled surface

3 2 S. Izumiya and N. Takeuchi Fig. is generally nonpositive. So the developable surface is a member of the special class of ruled surfaces. Therefore we have the natural question: How are singularities of developable surfaces different from those of general ruled surfaces? In this paper we give a classification of singularities of general ruled surface. A ruled surface in R 3 is (locally) the image of the map F (γ,δ) : I J R 3 defined by F (γ,δ) (t, u) =γ(t) +uδ(t), where γ : I R 3,δ: I S 2 are smooth mappings and I,J are open intervals. We assume that I is bounded. We call γ a base curve and δ a director curve. The straightlines u γ(t) + uδ(t) are called rulings. In order to describe the main result in this paper we need some preparation. Let f i :(N i,x i ) (P i,y i )(i =, 2) be C map germs. We say that f,f 2 are -equivalent if there exist diffeomorphism germs φ: (N,x ) (N 2,x 2 ) and ψ :(P,y ) (P 2,y 2 ) such that ψ f = f 2 φ. Let Cpr (I,R 3 S 2 ) be the space of smooth proper mappings (γ,δ): I R 3 S 2 equipped with Whitney C -topology, where I is an open interval. The following theorem is the main result in this paper which gives a generic answer to the above question. Theorem. There exists an open dense subset C pr (I,R 3 S 2 ) such that the germ of the ruled surface F (γ,δ) at any point (t,u ) is an immersion germ or -equivalent to the cross-cap for any (γ,δ). Here, the cross-cap is the map germ defined by (x,x 2 ) (x 2,x 2,x x 2 ). It is well known that any singular point for generic smooth mappings from a surface to R 3 is the cross-cap (cf. [, 5, 3, 9]). The set of ruled surfaces is a very small subset in the space of all C -mappings. The above theorem, however, asserts that the generic singularities of ruled surfaces are the same as those of C -mappings. We remark that the cross-cap is realized as a singularity of a ruled surface as follows: consider curves γ(t) =(t 2,, ) and δ(t) =(, / +t 2,t/ +t 2 ), then F (γ,δ) (t, u) is the cross-cap (cf. Fig. 2 below) which corresponds to the normal form. We can summarize the results of the above theorem as the following relations by referring to the previous results [3, 2, 5]: {Singularities of generic developable surfaces} {Singularities of generic ruled surfaces}, {Singularities of generic ruled surfaces} = {Singularities of generic C -mappings}.

4 Singularities of ruled surfaces in R Fig Fig. 3 One of the examples of ruled surfaces with cross-caps is the Plücker conoid which is given by γ(θ) =(,, 2 cos θ sin θ) and δ(θ) = (cos θ, sin θ, ) ( θ 2π) (cf. Fig. 3). We can also see a beautiful picture of the ruled surface at the home page of Banchoff [2]. In Section 2 we briefly review the classical theory of ruled surfaces. The idea of the proof of Theorem is that we may locally regard the ruled surface as a onedimensional unfolding of a map germ and apply the theory of unfoldings. In this case the parameter along rulings is considered to be the unfolding parameter. In Section 3 we prepare the general theory of unfoldings. The proof of Theorem isgivenin Section 4.

5 4 S. Izumiya and N. Takeuchi This is the first paper of the authors joint project entitled Geometry of ruled surfaces and line congruences. All manifolds and maps considered here are of class C unless otherwise stated. 2. Basic notions and a review of the classical theory We now present basic concepts and properties of ruled surfaces in R 3. The classical theory has been given in [6]. However, ruled surfaces are not so popular now, so that we review the classical framework. For the ruled surface F (γ,δ),ifδ is a constant vector v, then the ruled surface F (γ,v) is a generalized cylinder. Therefore, the ruled surface F (γ,δ) is said to be noncylindrical provided δ never vanishes. Thus the rulings are always changing directions on a noncylindrical ruled surface. It is clear that the set consisting of noncylindrical ruled surfaces is an open and dense subset in Cpr (I,R 3 S 2 ). Then we have the following lemma (cf. [6, lemmas 7 7, 7 8]). Lemma 2. () Let F (γ,δ) (t, u) be a noncylindrical ruled surface. Then there exists a smooth curve σ : I R 3 such that Image F (γ,δ) = Image F (σ,δ) and σ (t),δ (t) =, where, denotes the canonical inner product on R 3. The curve σ(t) is called the striction curve of F (γ,δ) (t, u). (2) The striction curve of a noncylindrical ruled surface F (γ,δ) (t, u) does not depend on the choice of the base curve γ. We can specify the place where the singularities of the ruled surface are located. Lemma 2 2. Let F (σ,δ) be a ruled surface with the striction curve σ. If x = F (σ,δ) (t,u ) is a singular point of the ruled surface F (σ,δ) then u =(i.e. x Image σ). Moreover, if σ (t ), then the ruling through σ(t ) is tangent to σ at t. Proof. We can calculate the partial derivative of F (σ,δ) as follows: Therefore we have F (σ,δ) t F (σ,δ) (t, u) =σ (t)+uδ (t), t F (σ,δ) (t, u) =δ(t). u F (σ,δ) u (t, u) =σ (t) δ(t)+uδ (t) δ(t), where denotes the vector product in R 3. Since δ(t) = δ(t),δ(t), we have δ (t),δ(t). By the condition that σ (t),δ (t) and the above, there exists a smooth function λ(t) such that σ (t) δ(t) =λ(t)δ (t). So we have F (σ,δ) t F (σ,δ) u (t, u) 2 = λ(t)δ (t)+uδ (t) δ(t) 2 = λ(t) 2 δ (t) 2 +2λ(t)u δ (t),δ (t) δ(t) + u 2 δ (t) δ(t) 2 =(λ(t) 2 + u 2 ) δ (t) 2. Suppose that x = F (σ,δ) (t,u ) is a singular point of the ruled surface F (σ,δ), then F (σ,δ) F (σ,δ) t u (t,u ) =. Since F (σ,δ) is noncylindrical, this means that u = λ(t )=.

6 Singularities of ruled surfaces in R Fig. 4 By Lemma 2 2, the singularities of a ruled surface are located on the striction curve. If we consider the cross-cap F (γ,δ) (t, u) = (t 2,u/ +t 2, ut/ +t 2 ), then γ (t) = (2t,, ) and δ (t) =(, t/ ( + t 2 ) 3, / ( + t 2 ) 3 ). By definition, γ(t) is the striction curve of F (γ,δ) (t, u) and the singular point is (,, ). We also consider the following examples. Example 2 3. Let γ : I R 3 and δ : I S 2 be curves given by γ(t) =(,,f(t)) and δ(t) = (cos t, sin t, ). Then the ruled surface F (γ,δ) (t, u) =(ucos t, u sin t, f(t)) is called a positive conoid. We can easily calculate that singularities of F (γ,δ) (t, u)isgiven by u =,f (t) = and the striction curve is γ(t). On the other hand, let g :(R 2, ) (R 3, ) be a smooth map germ. It has been known that the origin is the cross-cap if and only if there exists a local chart (x,x 2 ) around the origin such that the following conditions hold: g x (), g x 2 () = and det ( g 2 g (), (), 2 g x x x 2 x 2 2 ) (). By a direct calculation, F (γ,δ) (t, ) is the cross-cap if and only if f (t ) = and f (t ). The above condition means that t is a Morse singular point of f(t). Moreover, it is well-known that Morse functions are generic in the space of smooth functions. Therefore, this example confirms the assertion of the main theorem. One of the examples of positive conoids with cross-caps is the Plücker conoid which has been given in Section (cf. Fig. 3). Example 2 4. Consider the developable surface F (γ,δ) (t, u) =(u, 2t tu, t4 + 2 t3 u) with a cuspidal cross-cap. If we slightly perturb it into the ruled surface F(γ,δ)(t, ε u) =(u, 2t 2 3tu, 2 t4 + εt t3 u), we can easily show that the origin is a cross-cap. The situation is depicted in Fig. 4. The left picture is F (γ,δ) (t, u) and the right one is F(γ,δ).2 (t, u). 3. Unfoldings For the proof of Theorem, we need to prepare and review the theory of onedimensional unfoldings of map germs. The definition of r-dimensional unfolding of

7 6 S. Izumiya and N. Takeuchi f :(R n, ) (R p, ) (originally due to Thom) is a germ F :(R n R r, ) (R p R r, ) given by F (x, u) =(f(x, u),u), where f(x, u)isagermofr dimensional parameterized families of germs with f(x, ) = f (x). This definition depends on the coordinates of both spaces (R n R r, ) and (R p R r, ). We need the coordinate free definition of unfoldings [4]. Let f :(N,x ) (P, y ) be a map-germ between manifolds. An unfolding of f is a triple (F, i, j)ofmapgerms,wherei: (N,x ) (N,x ), j:(p, y ) (P,y ) are immersions and j is transverse to F, such that F i = j f and (i, f): N {(x,y) N P F (x )=j(y)} is a diffeomorphism germ. The dimension of (F, i, j) as an unfolding is dim N dim N. We can easily prove that the above two definitions are equivalent. Lemma 3. Let F :(R 2, ) (R 3, ) be a map germ with the components of the form F (t, u) = (F (t, u),f 2 (t, u),f 3 (t, u)). Suppose that ( F 3 / u)(, ). By the implicit function theorem, there exists a function germ g :(R, ) (R, ) with F 3 () = {(t, g(t)) t (R, )}. Let us consider immersion germs i: (R, ) (R 2, ) given by i(t) = (t, g(t)), j:(r 2, ) (R 3, ) given by j(y,y 2 ) = (y,y 2, ) and a map germ f :(R, ) (R 2, ) given by f(t) =(F (t, g(t)),f 2 (t, g(t))). Then the triple (F, i, j) is a one-dimensional unfolding of f. Proof. It is clear that F i = j f. Since ( F 3 / u)(, ), F is transverse to j. We can easily show that {(t, u, y,y 2 ) F (t, u) =j(y,y 2 )} = {(t, g(t),f (t, g(t)),f 2 (t, g(t)) t (R, )}. Since (i, f): (R, ) (R 2 R 2, ) is given by (i, f)(t) =(t, g(t),f (t, g(t)),f 2 (t, g(t))), it maps diffeomorphically onto the above set. This completes the proof. Since the cross-cap is a stable singularity of map germs (R 2, ) (R 3, ), we now discuss the stability of unfoldings. Let n be the local ring of function germs (R n, ) R and the unique maximal ideal is denoted by n. For a map germ f :(R n, ) (R p, ), we say that f is infinitesimally -stable if the following equality holds: f (n, p) =,..., f + f (p, p), x x n n where (n, p) denotes the n -module of map germs (R n, ) (R p, ) and f : (p, p) (n, p) isthepull back map given by f (h) =h f. It is known that an infinitesimally -stable map germ (R 2, ) (R 3, ) is an immersion germ or the cross-cap [, 5, 3, 9]. For map germs f,g: (R n, ) (R p, ), we say that they are -equivalent if there exists a diffeomorphism germ φ: (R n, ) (R n, ) such that f ( p ) n = φ g ( p ) n. The -equivalence is a equivalence relation among map germs. Let J k (n, p) bethek-jet space of map germ (R n, ) (R p, ). For any z = j k f() J k (n, p), we denote that k (z) ={j k g() g is -equivalent to f}. We call it a k -orbit since it is the orbit of a certain Lie group action. For any map germ f :(R n R r, ) (R p, ), we define a map germ j k f :(R n R r, ) J k (n, p) by j k f(x,u )=j k f u (x ), where f u (x) =f(x, u) and j k f u (x )=j k (f u (x + x ))(). We have the following Lemma (cf. [3]).

8 Singularities of ruled surfaces in R 3 7 Lemma 3 2. Under the same notation as the above, j k f is transverse to k (j k f ()) if and only if f (n, p) =,..., f f +f ( p ) (n, p)+ (x, ),..., f (x, ),e,...,e p x x n n u u r where e i =(,...,, i,,...,) R p. The following lemma is folklore. However, we cannot find any context on where the proof is explicitly written. So we give the proof here. Lemma 3 3. Let F :(R n R r, ) (R p R r, ) be an unfolding of f of the form F (x, u) =(f(x, u),u). If j k f is transverse to k (j k f ()) for sufficiently large k, then F is infinitesimally -stable. Proof. By Lemma 3 2 we may assume that f (n, p) =,..., f f +f ( p ) (n, p)+ (x, ),..., f (x, ),e,...,e p. x x n n u u r R We can show that f (n + r, p) =,..., f + f ( p ) (n + r, p) x x n n+r f + (x, ),..., f (x, ),e,...,e p + r (n + r, p). u u r r We now apply the Malgrange preparation theorem (cf. [3, p. 4]) as follows: consider M = (n + r, p) asan n+r -module of finite type. Then we have the quotient r -module M = (n + r, p)/ r (n + r, p). We also consider an n+r -submodule f N =,..., f + f ( p ) (n + r, p) x x n n+r of M. By the previous equality and the Malgrange preparation theorem, we have f M = N + (x, ),..., f (x, ),e,...,e p u u r This means that (n + r, p) = f,..., f +f ( p ) (n + r, p) x x n n+r f + (x, ),..., f (x, ),e,...,e p u u r For any ξ =(ξ,ξ 2 ) (n + r, p + r) = (n + r, p) (n + r, r), there exist λ i,η i n+r and µ i,ζ i r such that ξ = n λ i f x i + p η i f,i + r µ i f u i + r. p ζ i e i. r., R

9 8 S. Izumiya and N. Takeuchi Therefore, we have n F (ξ, ) = λ i + x i = n λ i F x i + p η i (f i, ) + r µ i F u i + r ( ) f µ i, + u i p η i (f i, ) + p ζ i (e i, ) p (ζ i µ i )(e i, ). Since ζ i µ i r, p (ζ i µ i )(e i, ) F (p + r, p + r). This means that F (ξ, ),..., F, F,..., F +F (p+r, p+r)+f ( p+r ) (n+r, p+r). x x n u u r n+r On the other hand, we have r F r ( ) f (,ξ 2 )= ξ 2,i ξ 2,i,. u i u i By the same argument as those above, we have F (,ξ 2 ),..., F, F,..., F +F (p+r, p+r)+f ( p+r ) (n+r, p+r). x x n u u r n+r Hence, we have F (n + r, p + r) =,..., F, F,..., F x x n u u r n+r +F (p + r, p + r)+f ( p+r ) (n + r, p + r). Applying the Malgrange preparation theorem once again, we have F (n + r, p + r) =,..., F, F,..., F + F (p + r, p + r). x x n u u r n+r 4. Generic classifications In this section we give the proof of Theorem. Since the infinitesimally -stable map germ (R 2, ) (R 3, ) is an immersion or the cross-cap, we now prove that the germ of the ruled surface F (γ,δ) at any point is infinitesimally -stable for generic (γ,δ). On the other hand, by the calculation of the proof of Lemma 2 2, the singular point of the ruled surface F (γ,δ) is given by the condition that rank (γ (t)+uδ (t),δ(t)) < 2 and it is equivalent to the condition that two vectors γ (t)+uδ (t), δ(t) are parallel. Since δ(t), rank (γ (t)+uδ (t),δ(t)). We now regard the parameter u (i.e. the parameter along the ruling) of the ruled surface as the parameter of a one-dimensional unfolding. For any (γ,δ): I R 3 S 2 with δ (t), we denote that γ(t) =(γ (t),γ 2 (t),γ 3 (t)) and δ(t) =(δ (t),δ 2 (t),δ 3 (t)), then we have the coordinate representation: F (γ,δ) (t, u) =(γ (t)+uδ (t),γ 2 (t)+uδ 2 (t),γ 3 (t)+uδ 3 (t)). For any fixed (t,u ) I J with δ 3 (t ), we define a non-empty open subset U 3 in I by U 3 = {t I δ 3 (t) }.

10 Singularities of ruled surfaces in R 3 9 We define a function g 3 (t) byg 3 (t) = (γ 3 (t) y )/δ 3 (t) for any t U 3, where y = γ 3 (t )+u δ(t ). Therefore, we have F (γ,δ) (t, u) =γ(t)+g 3 (t)δ(t)+(u g 3 (t))δ(t) =γ(t )+g 3 (T )δ(t )+Uδ(T ) for T = t, U = u g 3 (t). We denote the above map as F (γ,δ) (T,U). By Lemma 3, the map germ F (γ,δ) (T,U)at(t, ) is a one-dimensional unfolding of ˆπ 3 F (γ,δ) (T,) = (γ (T )+g 3 (T )δ (T ),γ 2 (T )+g 3 (T )δ 2 (T )), where ˆπ 3 : R 3 R 2 is the canonical projection given by ˆπ 3 (y,y 2,y 3 )=(y,y 2 ). The following lemma is the basis for the proof of Theorem. Lemma 4. Let W J k (, 2) be a submanifold. For any fixed map germ δ : I S 2 with δ (t) and any fixed point (t,u ) I J with δ 3 (t ), the set 3T W,(t,u ) = {γ j k ˆπ 3 F (γ,δ) is transverse to W at (t,u )} is a residual subset in C (I,R 3 ) {δ}. Here, we consider that C (I,R 3 S 2 )=C (I,R 3 ) C (I,S 2 ) and relative topology on C (I,R 3 ) {δ}. For the proof of Lemma 4, we need to apply the following fundamental transversality lemma of Thom (cf. [5, p. 53, lemma 4 6]). Lemma 4 2. Let X, B and Y be C -manifolds. Let j : B C (X, Y ) be a mapping (not necessarily continuous) and define Φ: X B Y by Φ(x, b) = j(b)(x). Assume that Φ is smooth and transverse to the submanifold W of Y. Then the set {b B j(b) is transverse to W } is dense in B. Proof of Lemma 4. Let {K j } j= be the countable set of open covering of W such that each closure K j is compact. We define the following set 3T W,(t,u ),K j = { γ j k ˆπ 3 F (γ,δ) is transverse to W with j k ˆπ 3 F (γ,δ) (t,u ) K } j. We now prove that 3 T W,(t,u ),K j is an open subset. For the purpose, we consider the following mapping ĵ k : C (U 3, R 3 ) C (U 3 J, J k (, 2)) defined by ĵ k (γ) =j k ˆπ 3 F (γ,δ). It is clear that the mapping ĵ k is continuous. We also define a subset O W,Kj = {g C (U 3 J, J k (, 2)) g is transverse to W at (t,u ) with g(t,u ) K j }, then it is open (cf. [5]). Since the restriction map res U3 : C (I,R 3 ) C (U 3, R 3 ) is continuous, 3 T W,(t,u ),K j = (res U3 ) (ĵ k ) (O W,Kj ) is open. If we show that 3T W,(t,u ),K j is dense subset in C (I,R 3 ) {δ}, then T W,(t,u ) = j= T W,(t,u ),K j is a residual subset. Since res U3 is surjective, it is enough to show that T W,(t,u ),K j,u 3 = { γ C (U 3, R 3 ) j k ˆπ 3 F (γ,δ) is transverse to W at (t,u ) with j k ˆπ 3 F (γ,δ) (t,u ) K } j is a dense subset in C (U 3, R 3 ).

11 S. Izumiya and N. Takeuchi For any γ C (U 3, R 3 ) and p = (p,p 2 ) P (, 2; k), we define a mapping f (γ,p) : U 3 J R 2 by f (γ,p) (t, u) =(γ (t)+p (t)+g 3 (t)δ (t)+uδ (t),γ 2 (t)+p 2 (t)+g 3 (t)δ 2 (t)+uδ 2 (t)), where P (, 2; k) denotes the space of pairs of polynomials (p,p 2 ) with degrees at most k without constant terms. We also define a mapping Φ: U 3 J P (, 2; k) J k (, 2) by Φ(t, u, (p,p 2 )) = j k f (γ,p) (t, u) =j k f (γ,p),u (t), where f (γ,p),u (t) =f (γ,p) (t, u). We may consider that P (, 2; k) is Euclidian space R N. It is easy to show that Φ is a submersion, so that it is transverse to W. By Lemma 4 2, {p =(p,p 2 ) P (, 2; k) Φ (p,p 2) is transverse to W at (t,u ) with Φ (p,p 2)(t,u ) K j } is dense in P (, 2, ; k). Hence, we can find (p,p 2 ), (p,p 2 ) 2, (p,p 2 ) 3,... in P (, 2; k) converging to (, ) so that Φ (p,p 2) i is transverse to W on K j. Since lim i (γ + ((p,p 2 ) i, )) = γ in C (U 3, R 3 ),T W,(t,u ),K j,u 3 is dense in C (U 3, R 3 ). We remark that j T W,(t,u ) (j =, 2) can also be defined for (t,u ) I J with δ j (t ) and the same assertion for j T W,(t,u ) as the above holds. Proof of Theorem. Let i be the -orbit with codimension i in J k (, 2) for sufficiently large k. We also denote that Σ(, 2) = i 3 i J k (, 2). It has been known that Σ(, 2) is a semi-algebraic subset in J k (, 2) with codimension greater than 2. Therefore we have the canonical stratification { i } m of Σ(, 2) with codim i > 2. For any (t,u ) with δ 3 (t ), we denote that 3 T Σ(,2),(t,u ) = m T i,(t,u ). Since 3 T i,(t,u ) and 3 T Σ(,2),(t,u ) are residual subsets in Cpr (I,R 3 S 2 ), 3 (t,u ) = 2 3T i,(t,u ) TΣ(,2),(t,u ) is also a residual subset in Cpr (I,R 3 S 2 ). By the remark after the proof of Lemma 4, j (t,u ) (j =, 2) are also residual subsets in Cpr (I,R 3 S 2 ) respectively. Therefore, for any fixed (t,u ) I J, there exists a residual subset (t,u ) Cpr (I,R 3 S 2 ) such that the map germ F (γ,δ) at (t,u ) is an infinitesimally -stable map germ for any (γ,δ) by Lemma 3 3. Since the infinitesimally -stable map germ R 2 R 3 is an immersion or the cross-cap and the singularities of F (γ,δ) are located on the striction curve, there exists an open neighbourhood U t I of t such that F (γ,δ) is an immersion on U t I {(t,u )}. Since Ī is compact, we can extend (γ,δ) slightly on an open interval I Ī and there exist finitely many U t i (i =,...,l) such open subsets as the above with I = l U t i. Then = l (t i,u i) is a residual subset of Cpr (I,R 3 S 2 ). It is clear that the germ F (γ,δ) at any point (t, u) I J is an immersion or the cross-cap for any (γ,δ). It is easy to show that the mapping F : C (I,R 3 S 2 ) C (I J, R 3 ) defined by F (γ,δ) =F (γ,δ) is continuous. Since the cross-cap is the stable singularities of map germs (R 2, ) (R 3, ), the set = {f C (I J, R 3 ) f is an immersion or the cross-cap at any point I J} is an open subset. Therefore, = F ( ) is an open subset of Cpr (I,R 3 S 2 ). This completes the proof of Theorem. REFERENCES [] V. I. Arnol d, S. M. Gusein-Zade and A. N. Varchenko. Singularities of differentiable maps, vol. I (Birkhäuser, 986). [2] T. Banchoff. Front cover of Notices of of A.M.S. ( vol. 44, no. 3 (997).

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