Geometriae Dedicata 54: 323-332, 1995. © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

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The Geometry of Surfaces in 4-Space from a Contact Viewpoint DIRCE KIYOMI HAYASHIDA MOCHIDA 1, MARIA DEL CARMEN ROMERO FUSTER 2. and MARIA APARECIDA SOARES RUAS 3 ** 1Departamento de Matemdtica, Universidade Federal de Sao Carlos, 13560-905, Sao Carlos, SP, Brazil. e-mail address: ddhm@power, ufscar.br 2Depto. de Geometria y Topologia, Universidad de Valencia, 46100 Valencia, Spain. e-mail address: [email protected] 3Instituto de Ci~ncias Matem6ticas de S~o Carlos, Universidade de Sao Paulo, Departamento de Matemdtica, 13560-970, S~o Carlos, SP, Brazil. e-mail address: maasruas@ icmsc.usp.br (Received: 2 December 1993; revised version: 23 March 1994)

Abstract. We study the geometry of the surfaces embedded in ~ 4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any genetic convexly embedded 2-sphere in ~ 4 has inflection points. Mathematics Subject Classifications (1991): Primary 58C27; Secondary 53A05.

1. Introduction

The geometry of compact surfaces in 4-space has been studied by several authors (e.g. [1], [2], [3], [4], [6], [9], [14]). Little described in ([6], 1969) the local secondorder invariants of these surfaces by using singularity theory techniques in the way developed by Feldman ([2], [3]). On the other hand, Porteous ([10], 1971) related the singularities of the distance-squared functions with the local geometry of generic submanifolds of Euclidean space. This study was pushed forward, in particular for surfaces in 4-space, by Montaldi ([9], 1983), who described the generic contacts of them with k-spheres in IR4. We obtain in this work several results on the geometry of surfaces in ~4 based on the analysis of their generic contacts with hyperplanes. We start by given a geometrical characterization to the singularities of the height functions on such manifolds. The inflection points (in the sense of Little [6]) are shown to be the umbilics of these functions. As a consequence of this setting we prove that, generically, any convexly embedded 2-sphere in IR4 must have inflection points. * The research of the second author was partially supported by Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico (CNPq), Brazil. ** The research of the third author was partially supported by Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico (CNPq), Brazil.

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Mond studied in [8] the generic contacts of surfaces with straight lines in 4space. Some of his conclusions are related to ours. For instance, we also get that the parabolic curve of M has transverse self-intersections at inflection points, but we observe that isolated inflection points, corresponding to umbilic points of height functions on M, may also appear off this curve. In fact, we show that these are the only umbilics generically appearing on the convexly embedded surfaces On the other hand, we identify Mond's curve of non-twisting asymptotic directions with the curve of cusps singularities of height functions. 2. The Second-Order Invariants of M

We take [6] as a basic reference for this section. Given m E M, consider the unit circle in TI(m)f(M) parametrized by the angle O E [0, 27r]. Denote by 70 the curve obtained by intersecting f ( M ) with the hyperplane at f(m) composed by the direct sum of the normal plane Nf(m)f(M) and the straight line in the tangent direction represented by 0. Such a curve is called the normal section off(M) in the direction 0. The curvature vector ~(0) of To in f(m) lies in NI(m)f(M ). Varying 0 from 0 to 27r, this vector describes an ellipse in Ny(m)f(M), called the curvature ellipse of f(M) at f(m). This ellipse may degenerate on a radial segment of straight line, in which case f(ra) is known as an inflection point of the surface. The inflection point is of real type when f(m) belongs to the curvature elipse, and of imaginary type when it does not. An inflection point is flat when f(ra) is an end point of the curvature elipse. The torsion 7"oof 7o at f(m) is called the normal torsion of f(M) in the direction 0 at f(ra). A direction 0 in Tf(m)f(M ) for which 0~1/00 and rl(O) are parallel is an asymptotic direction. The second fundamental form of f ( M ) is characterized by two quadratic forms. Their functional coefficients will be denoted by (a, b, e ) and ( e, f, g) respectively. We then have the following functions associated to them:

A(m) = ldet

a2b c o e2fgo (m) o a 2bc o e 2f g

k(m) = (ac - b2 + eg - f2)(m). (Gaussian curvature of M ) and the matrix

a(m)--- ( a( mfb) 'gce) By identifying

f(m) with the origin of Nf(m)f(M) it can be shown that:

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(a) A ( m ) < 0 =~ f ( m ) lies outside the curvature ellipse (such a point is said to be a hyperbolic point of M); (b) A ( m ) > 0 =~ f ( m ) lies inside the curvature ellipse (elliptic point); (c) A(ra) = 0 =~ f(ra) lies on the curvature ellipse (parabolic point). A more detailed study of this case allows us to distinguish among the following possibilities:

A ( m ) = O, k(m) > 0 =~ f ( m ) is an inflection point of imaginary type rank a ( m ) = 2 =~ ellipse is non-degenerate A ( m ) = O, k(m) < 0 and rank a ( m ) = 1 =~ f ( m ) is an inflection point of real type A ( m ) = O, k(ra) = 0 ~ f ( m ) is an inflection point of flat type. It can be shown that for an open and dense set of embeddings of M in 11~4, A -1 (0) fq k - l ( 0 ) = ~. It was proved in [8] that there are exactly two asymptotic directions at a hyperbolic point and just one asymptotic direction at a parabolic point, unless it is an inflection point, in which case all the directions are asymptotic. 3. The Canal 3-Manifold of a Surface in ~ 4 The canal 3-manifold of a surface in ]R4 is defined as C M = { f ( m ) +ev E ]~4: m E M and v is an unit normal vector to f ( M ) at f ( m ) } , here e is a sufficiently small positive real number chosen such that C M is embedded in ~4. We denote by f the natural embedding of C M in ]~4 and by (m, v) the point f ( m ) + ev E CM. From Looijenga's theorem ([7]), it follows that there is a residual subset of embeddings f: M 2 ~ R 4, for which the family of height functions: A(f): M × S 3 "--+]R

(m, v ) ~ (f(m), v ) = fv(m) is locally stable as a family of functions on M with parameters o n S 3. Moreover the corresponding family )~(f) on the canal manifold is also generic. In fact the singularities of )~(f) and )~(f) are tightly related ([13]). These may be, for a generic f , of one of the following types: Morse (A1) , fold (A2), cusp (A3), swallowtail (A4) and elliptic or hyperbolic umbilic (Dff). Moreover, the singularities of the normal Gauss map, P: C M --~ S 3 (also called generalized Gauss map on M ) can be described in terms of those as follows: LEMMA 3.1 ([11], [13]). Given a critical point (m, v) E C M of the height function fv (or equivalently, given a critical point m E M of fv), (a) m is a nondegenerate critical point of fv ¢:> (ra, v) is a regular point of F, or equivalently, (a 1) m is a degenerate critical point of f~ ¢~ (m, v) is a singular point of P.

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Let/Co: C M --+ 1R be the Gaussian curvature function on C M . The parabolic set, E l l ( 0 ) , of C M is the singular set of lP. It can be shown [12] that for a generic embedding of M, E l l ( 0 ) is a regular surface except by a finite number of points (m, v) which are singularities of type p2,0 of F, or equivalently, umbilic points (Dff) of)~. Let ~: C M --+ M be the natural projection of C M onto M, i.e. ~(m, v) = m. The image of the set of parabolic points K:~-I(0) by ~ is the set A _< 0. More precisely: LEMMA 3.2. (1) If A ( m ) > 0, then m is a non-degenerate critical point of .Iv, Vv E N s ( m ) f ( M ). (2)/f A(m) < 0, then there are exactly two vectors bl, b2 E Ny(m)f(M), such that m is a degenerate critical point of fbi. (3) If A ( m ) = O, then there is a unique vector b C N y ( m ) f ( M ) such that m is a degenerate critical point of fb. Proof Let f: (JR2, O) --+ (]1~4, O)

(x, y) ~+ (x, y, fl(x, y), f2(x, y)) be the local expression of the embedding in Monge's form and fv: (~2, 0 ) ~ + (~, 0) (x, y)

~-+ fv(x, y) = VlX + v2y ~- v3fl(x, y) W v4f2(x, y)

the height function in v-direction, where v = (Vl, '02, ?33, "04) E ]t~4. If (0, 0) is a critical point of the height function f,,, then v = (0, 0, v3, v4) and the determinant of the Hessian matrix of fv at (0, 0) is given by: det 7-/(fv)(0, 0) -- ( a c - b 2 ) v 2 + (ag + c e - 2bf)v3v4 + ( e g - f2)v2, where (a, b, e), (e, f, g) are the coefficients of the second fundamental form of M at (0, 0). Now, A = (ac - b2)(eg - f2) _ ¼(ag + ce - 2bf) 2

and the equation det 7-/(fv)(0, 0) = 0 has two, one or zero solutions (v3, v4) as A < 0, A = 0 or A > 0, respectively. [] Remark 3.1. When m is a degenerate critical point of fb, the hyperplane 11b, orthogonal to b has a higher order contact with f ( M ) at f ( m ) . Therefore, by analogy with curves in R3, we shall say that b is a binormal vector of f ( M ) at f ( m ) and Hb an osculating hyperplane.

Let ~ be the restriction of ~ to the surface/C~1(0) - p2(p), and denote by M_ = {m C M : A ( m ) < O} and B = {(m, v) C/C~-l(o):m E M_}.

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PROPOSITION 3.1. (i) ~IB: B --+ M_ is a local diffeomorphism, more precisely,

it is a double covering. (ii) A ( m ) = 0 and m is not an inflection point ¢* there is v C S 3 such that (m, v) is a singular point (fold) of ~. Proof We choose coordinates for C M such that m = (0, 0) and v = (0, 0, 0, 1). Now, it is sufficient to notice that if v is a degenerate direction then det 7-[(fv)(O, O) = ( a c - b2)v 2 + (ag + ce - 2by)v3 + ( e g - f2) = ]~c(m, V3) : O.

Then: (i) v3 = 0 is a simple root of/Co(m, v3) = 0 ¢~ (OEc/Ov3)(O, 0) ¢ 0 ¢¢, ~ is a local diffeomorphism. (ii) v3 = 0 is a double root of/~c(ra, v3) = 0 ~ (OE_c/Ov3)(O, O) = 0 mad (OZlC~/Ov2)(O, 0) ~ 0 ~ (m, v ) i s a fold point for ~. At each point of/C~ -1 (0) - E2(F) there is an unique principal direction of zero curvature for C M . This direction is tangent to the surface/C[ 1(0) on a curve made of points of type E l'l(F). This curve is in turn tangent to a zero principal direction of curvature at isolated points which are of type El'l'l(I'). (See [12].) PROPOSITION 3.2. The image of the zero principal directions of curvature in /C~-1(0) - Ez(F) under ~ are asymptotic directions on M (here we use the

identification M = f ( M ) ). Proof The following expression for the curvature vector ~7(0) is given in [6, p. 265]: Z](0) :

(l(a ~,2

- c) cos 20 + b sin 20)e3

+(½(e - g) cos 20 + f sin 20)e4 + 7-/, where 7-/ = 2!(a -t- c)e3 -l- l(e + g)e4, is the mean curvature vector. Now we can choose local coordinates for M such that ~(m) =

0 0 1

"

This choice will imply that el = ( 1, 0, 0, 0) C T(m,v)CM is the zero curvature direction, and T~(rn, v).el = el E TmM. Then, it follows easily that ~(0) and (On/O0)(O) are parallel. 4. Geometric Characterization of the singularities of the Height Functions on M We have seen that a height function f~: M ~ ~ has a degenerate singularity at m if and only if v is a binormal vector of f ( M ) at f ( m ) . In this section we characterize the type of degenerate singularities that occur generically.

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Denote by 7 the normal section of M tangent to the asymptotic direction 0 at f ( m ) associated to the binormal vector v, and let X be the curve made of points of type N1,1 (i.e. cusps and swallowtails of I'). THEOREM 4.1. For m E M such that A(m) < 0: m is a fold singularity of fv ¢¢" 7 has non-vanishing normal torsion at m. Now, if'/has vanishing normal torsion at m, m E ((X) and we have that: (i) m is a cusp singularity of f~ ¢~ 0 is transversal to ((X). (ii) m is a swallowtail singularity of fv ¢¢" 0 is tangent to ((X) with contact of order I at m.

Proof As before, we can choose local coordinates such that f is given in Monge's form, and the degenerate direction v is (0, 0, 0, 1). That is, N2, 0 ~ I~4, 0 (x, y) ~

(x, y, fl(x, y), f2(x, y))

with

f l ( x , y)

= ax 2

+ 2bxy +

cy 2

-]- Mix 3 +

3M2x2y

+ ""

fz(x, y) = y2 + plx 3 + 3P2x2y + 3P3xy2 + p4y3 + QlX 4 + . . . . Then, the Gaussian curvature/Co is given by: N 2 × N , 0 E% N, 0 (x, y, v3) '

) K:c(x, y, v 3 ) = A o ( x , y)v 2 + A I ( x , y ) v 3 + A 2 ( x ,

y)

where do(x, y ) = det M ( q l ) = flxx(x, y).flyy(x, y ) - f 2 y ( x , y)

AI(x, y)= fl (x, y).f2,(x, y)+ fl (x,

y)

-2flay(x, y).f2~u(x, y) A2(x, y ) = det M ( q 2 ) = f2~(x, y).f2~y(x, y ) - f2~,y(x, y). The zero curvature direction of C M at (m, v) in these coordinates is the xaxis. Then T~(p, v).el = el = 0 (Proposition 3.2). The normal section 7 can be parametrized by

(a, o)

o)

s ~+ "7(8) = (S, O, a s 2 - 1 - . . . ,

/°1 s3 + ' " ")

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and it follows that 7 has non-zero torsion ~ P1 76 0 ¢* m is a fold point of f~. Now, if P1 = 0, (m, v) C EI'I(I?). Then (i) m is acusp point o f f v ¢:~ (ra, v) E EI'I'°(F) ¢¢' the zero curvature direction is tangent to/~[1 (0) and transverse to the curve EI,I,°(F); (ii) m is a swallowtail singularity of f~ ¢¢. (m, v) E Eld'l'°(F) ¢¢" the zero principal direction is tangent to E 1,1,0(i,), with first-order contact. Since ~: B ~ M_ is a local diffeomorphism, O is transversal to ((El'l,°(I')) in (i) and tangent to ((EI'I'°(F)) with first-order contact in (ii). An immediate consequence of this theorem is that what Mond [8] calls curves of non-twisting asymptotic directions are precisely the curves made of points at which the height function in one of the binormal directions has a singularity of codimension at least 2 (cusp or swallowtail in the genetic case under consideration). The characterization of the singularities of the height functions on the curve A -1 (0) is given by the following: THEOREM 4.2. (i) If A ( m ) = O, but m is not an inflection point of M ; then m is either a fold or a cusp singularity of f~ and: • m is a fold singularity of f~ ¢~ 0 is transversal to the curve A - l ( 0 ) of parabolic points of f ( M ) . • m is a cusp singularityoff~ ¢¢, 0 is tangentto A - l ( 0 ) . (ii) m is an umbilic of f~ ¢~ m is an inflection point of M. Moreover, • m is a normal crossing point of A -1 (0) ¢¢, m is an inflection point of real type; • m is an isolated point of A - l ( 0 ) ¢~ m is an inflection point of imaginary type. Proof (i) Observe first that the curves of M composed of points having a singularity more degenerate than a fold cannot meet the curve A - l ( 0 ) in a swallowtail point. This follows from the Thom transversality theorem by observing that for a point (x, y) to be a swallowtail point of fv, the 4-jet extension j 4 f : M × S 3 ~ j 4 ( m , ~ ) must meet an algebraic variety of codimension 5 in j 4 ( M , ~). Now, the fact that a point belongs to the curve A - l ( 0 ) also amounts to saying that the image o f j 4 f at this point cuts another algebraic variety of codimension 1. Thus to have both conditions at the same time, the map j 4 f needs to meet the intersection of both varieties, which has codimension 6 in j 4 ( M , ~). But this can be avoided for a genetic embedding f. With the same choice of coordinates as in Theorem 4.1, we have: /Co(x, y, v3) = Ao(x, y)v~ + A l ( x , y)v3 + A2(x, y),

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Z @ el

bl

c]

dl

Fig. 1. (a) and (c) show a hyperbolic umbilic, (b) and (d) show an elliptic umbilic. The dotted lines represent the curve of cusps for the height function.

where A0(x, y ) =

- 4 b 2 + 12[(eMI - 2bMz)x + (eM2 - 2bM3)y] + . . . , A0(0, 0) ¢ 0

a l ( x , y) = 12[(M1 + c]91 -- 2bP2)x + (11//2 + cP2 - 2bP3)y] + . . . , d2(x, y) = 12(Fix + P2Y) + [(24Q1 + 36(P1P3 - P22)]x2 +[48Q2 + 36(P1P4 - P2P3)]xy +[24Q3 + 36(P2P4 - p2)]y2 +[40R1 + 72(P1Q3 + P3Q1 - 2P2Q2)]x 3 + . . . Now, we can see easily that the discriminant set {(x, y)/3v3:~Cc(x, y, v 3 ) = 0

and

(01Cc/Ov3)(x, Y, v 3 ) = O}

isA0(x, y).Az(x, y ) - ~Al(X I 2 , y), which is exactly the set A = 0. Then: (a) ra is a fold point of f~ ¢¢, (OA2/Ox)(O, 0) = 12/°1 ~ 0 ¢¢, the asymptotic direction el is transversal to the curve A = 0. (b) At a cusp point, Ax(0, 0) = -4b2P1 = 0, Au(0 , 0) = -4b2P2 # O, and thus the asymptotic direction el is tangent to the curve A = 0. WemustremarkthatifAxx(0, 0 ) = -3/414b2(2Q1-3P~)+3(M1-2bPz) z] 7~ 0, the contact of the asymptotic direction with the curve A = 0 is of order 1. Although this condition is verified for a residual set of embeddings, it does not follow from the conditions defining a cusp point. (ii) m is an umbilic point of f~ ~ rank ~ = 1 ¢, m is an inflection point of M. Moreover, to see the geometry of A -1 (0) at an inflection point, we compute (using local coordinates) the Hessian matrix of the function A: M --+ 11~. We have: det 7-/(A)(0, 0) = pk(0, 0) where p is a positive real number. Thus, whenk(0, 0) > 0, A - I ( O ) r e d u c e s t o a p o i n t ; w h e n k ( 0 , 0) < 0, A - t ( o ) is a curve with normal crossing at m. Figure 1 shows the projection ~ restricted to the parabolic set of )U~-1( 0 ) in a neighborhood of an umbilic of CM. []

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Remark 4.1. Each one of the connected components of A - l ( 0 ) can be one of the following types: (1) embedded curve; (2) immersed curve with a finite number of transverse self-intersections, and (3) isolated point. COROLLARY 4.1. Given a generic embedding f of M in ~4 with no inflection points, the set of parabolic points, A - 1(0) is a disjoint union of circles. COROLLARY 4.2. Given a generic embedding f of M in ]~4 with no inflection

points then: (1) Ho(M_) = Ho(/~-l(O)) (2) H 0 ( A - I ( 0 ) ) = H0(K~cl(0)) q- g(/C:l(0)), where 9 denotes the genus and

Hj the jth homology group with integer coefficients. Proof In this case, )UZ-I(0) is a closed surface, and each one of its connected components projects onto a connected component of M_. Moreover, each component with genus g of)U~-l(0) gives rises to g + 1 components of A-~(0). We say that a surface M in ]t~4 is convex provided it lies entirely on the boundary of its convex hull (minimum convex set of ~ 4 that contains M). The analysis above leds to the following results on convex surfaces in IR4. COROLLARY 4.3. Let f be a generic embedding of M in IR4. Then, M is locally convex if and only if A <_0 and A - 1 (0) consists of isolated inflection points of M. Proof. M is not locally convex at a point p such that A(p) > 0. In fact, in this case all the height functions have a non-degenerate saddle at p. On the other hand, it is easy to see that if A(p) < 0, M has support hyperplanes. The case A --- 0 follows from the analysis of the local models discussed in Theorem 4.2. Little proved in [6] that surfaces with non-zero Euler characteristic embedded in ~ 4 have inflection points or points with zero mean curvature. If the surface is contained in 5'3 , he observes that the mean curvature has to be non-zero, hence the embedding must have inflection points. We prove below that this result remains true for convex generic embeddings of 2-spheres. COROLLARY 4.4. A generic convex embedding f of 5"2 in k1~4 has inflection

points. Proof. Suppose that f has no inflection points. Then, from Corollary 4.2, Ho(M-) = H0(/C;-I(0)). Since M is convex, it follows from Corollary 4.3 and the hypothesis that the set A - 1(0) is empty. Then,/C~- 1(0) is diffeomorphic to two disjoint copies of 5'2, contradicting Corollary 4.2(1).

Acknowledgment We thank James Montaldi and Brasil T. Leme for valuable suggestions.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Bishop, R. L. and Menn, M.: Generic surfaces in E 4, Michigan Math. J. 22 (1975), 117-127. Feldman, E. A.: Geometry of immersions I, Trans. Amer. Math. Soc. 120 (1965), 185-224. Feldman, E. A.: Geometry of immersions II, Trans. Amer. Math. Soc. 125 (1966), 181-315. Forsyth, A. R.: Geometry of Four Dimensions, Cambridge University Press. Golubitsky, M. and Guillemin, V.: Stable Mappings and their Singularities, Springer-Verlag, New York, 1973. Little, J. A.: On singularities of submanifolds of higher dimensional Euclidean space. Ann. Mat. PuraAppL (Ser. 4A) 83 (1969), 261-336. Looijenga, E. J. N.: Structural stability of smooth families of C~-functions, Thesis, University of Amsterdam, 1974. Mond, D.: Thesis, Liverpool, 1982. Montaldi, J. A.: Contact with applications to submanifolds, Thesis, Liverpool, 1983. Porteous, I. R.: The normal singularities of submanifolds, J. Differential Geom. 5 (1971), 543564. Rodrigues, L.: Geometria das Subvariedades, Monografias de Matem~itica, No. 26, IMPA, Rio de Janeiro. Romero Fuster, M. C.: Sphere stratifications and the Gauss map, Proc. Royal Society of Edinburgh, 95 A (1983), 115-136. Smith, M. I. N.: Curvature, singularities and projections of smooth maps, Thesis, Durham, 1971. Weiner, J.: The Gauss map for surfaces in 4-space, Math. Ann. 269 (1984), 541-560.