REFERENCE IC/88/392
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
THE LOCAL STRUCTURE OF A 2-CODIMENSIONAL CONFORMALLY FLAT SUBMANIFOLD IN AN EUCLIDEAN SPACE lRn+2
G. Zafindratafa
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION
IC/88/392 ABSTRACT
International Atomic Energy Agency and Since 1917, many mathematicians studied conformally flat submanifolds (e.g. E. Cartan, B.Y. Chen, J.M. Morvan, L. Verstraelen, M. do Carmo, N. Kuiper, etc.). United Nations Educational Scientific and Cultural Organization In 1917, E. Cartan showed with his own method that the second fundamental form INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS of a conforraally flat hypersurface of IR™+1 admits an eigenvalue of multiplicity > n — 1. In order to generalize such a result, in 1972, B.Y. Chen introduced the notion of quasiumbilical submanifold. A submanifold M of codimensjon JV in 1R"+W is quasiumbilical if and only if there exists, locally at each point of M, an orthonormal frame field {d,...,£jv} of the normal space so that the Weingarten tensor of each £„ possesses an eigenvalue of multiplicity > n — 1. THE LOCAL STRUCTURE OF A 2-CODIMENSIONAL CONFORMALLY FLAT SUBMANIFOLD The first purpose of this work is to resolve the following first problem: IN AN EUCLIDEAN SPACE JRn+;1 * Is there any equivalent definition of quasiumbilictty which does not use either any local frame field on the normal bundle or any particular sections on the tangent bundle? B.Y. Chen and K. Yano proved in 1972 that a quasiumbilical submanifold of G. Zafmdratafa, ** codimension N > 1 in JR"+" is conformally flat; moreover in the case of codimension International Centre for Theoretical Physics, Trieste, Italy. JV = 1, the notion of quasiumbilicity is equivalent to the conformal flatness. J.D. Moore and J.M. Morvan improved this last equivalence in 1978 by proving that, if 1 < N < inf (4,n — 3), a N-codimensional submanifold of IR"+A' is conformally flat if and only if it is quasiumbilical. But two years before, B.Y. Chen and L. Verstraelen demonstrated that any N-codimensional conformally flat submanifold of IRrl+w (1 < iV < n — 3) with a null normal curvature is quasiumbilical. In 1985, using these statements, M. do Carmo and his coworkers determined the local geometry of a conformally flat hypersurface of K.n+1. The second purpose of this work is to generalize such a M. do Carmo's result in codimension 2 in IR"+2. One of our MIRAMARE - TRIESTE tools wili be the notion of focal sets, as we did in the resolution of our first problem. December 1988
' To be submitted for publication. " Permanent address: E.E.S.S., Universite de Fianarantsoa, 301 Fianarantsoa, Madagascar. 0. INTRODUCTION Is there any equivalent definition ofif quastumbilicityquasiumbilicity which does not use either. any local frame field on the normal bundle or anyny particularparticular sectionssections onon thetht tangenttangent bundle'! 0.1 Since 1917, many mathematicians as E. Cartan, B.Y, Chen, J.M. Morvan, L. Ver- In the first two sections of this work, we recall some basic definitions. We answer straelen, M. do Carmo, N. Kuiper and some others studied conformally flat submanifolds. our first question in Theorems 1 and 1-bis, and in Sec. 3, by means of focal sets. In order A submanifold in an Euclidean space is conformally flat if, when it is endowed to really visualize this concept, we will determine in Sec. 4 the focal set of any element with the induced metric, say < -,- >, each point belongs to an open neighbourhood U in the class CP* — Q^J, for n > 5. This enables us to resolve (see Theorems 2 to 5 and which possesses coordinates i1,... xn such that: Sec.5) the following problem:
How to find the local geometry of a conformally flat submanifold of codimension <-,->= e"{dxl ® dx1 + ... + dx" ® dx") 2 in R"+2, as M. do Carmo did in [DDM] for hypersurfaces in the class CP^l where p € C°°[U), and n = dimM. Pr.J.M. Morvan purposed such a problem to the author in 1987. (It must be E. Cartan first proved with his own method in 1917 [CA] that the second funda- mentioned that the author has already written in 1986 in his thesis ([ZA]} all the results mental form of a conformally flat hypersurface on E."+1 can be represented by a matrix stated in Sees. 3 and 4, in collaboration with Pr. J.M. Morvan).
0.2 Statements of the results A 0 (for the definitions and the notations, see Sees. 1,2,3) 0 •. Theorem 1 Let Me ST^1 (1 < N < n) and p be a point in M where the first canonical normal vector field a vanishes. Secondly, in 1949 [KN], N. Kuiper demonstrated that a simply connected compact conformally flat hypersurface of 1R"+1 is nothing but an n-sphere Sn. 36 years after, M. Then M is quasiumbilical at p if and only if there exists an affine orthogonal +W do Carmo and his coworkers improved in [DDM] such a Kuiper's result by removing the JV-frame {f(p)\qi,. .-«„,. ..,«jv} in JR" such that: simply connected condition. Some other mathematicians attempted to generalize those {!) 9i> • • • Qa are focal points of Q,, with multiplicity one, for a certain a, works and to determine the class CPj? of all conformally flat submanifolds of codimension 1 < a < N, JV (JV > 2) in K"+w. Unfortunately, they could not succeed in a complete classification. It is important to underline that almost all these people used a common approach: they (2) there is no focal point in each straight line passing by f(p) and
(1) gi,...,<7n are focal points of [M,p) belonging to Pp n Qt, with multiplicity In 1978, J.M. Morvan and J.D. Moore proved in [MM] that if the codimension N re — 1 or n; satisfies 1 < N < inf(4,n-3), then QO^ = CPJ?. But two years before, B.Y. Chen and L.
Verstraelen obtained the same equality in [CV] if the normal curvature tensor is null and (2) <7a+i,.. .qp are focal points of (M,p) in Qp with multiplicity one, so that, for if, in addition: 1 < JV < n - 3. each "y, a < T < /?, the line [q^\ passing by /(p) and qn does not intersect Pp;
In the light of all these statements, Pr. A. Lichnerowicz and Pr. J. Martinet ask, (3) For any S £ {[} + 1,..., N}, the line [qs] passing by f(p) and qt, contains no in 1979, the following first question to Pr. J.M. Morvan: focal point. Theorem 2 Let M be a two codimensional conformally flat submanifold in the Eu- Then: clidean space JR"+2 (n > 5) without umbiiic points, so that: (1) the normal connection is flat;
(i) Vp € M, d\m{EL)p = 1; (2) M is foliated by umbilical (n - l)-submanifolds in IR"+2; (ii) Vp e M, ap + 0;
(iii) The normal direction £i = TAT is not cylindrical. (3) M is locally the envelope of a one parameter family (^""'(t of (n —1)-spheres 1 2 Then the second principal normal space E2 of M is everywhere reduced to zero. in B."" " with radii r = r(t) = T4TT and with centers at the focal points c — c[t) = f + vAp In particular, M can be immersed as a conformally flat hypersurface of a totally geodesic of multiplicity n\ hyperplane of IR"+2. Furthermore:
(4) At each point p £ M, the focal set 7V is the union of two intersecting lines (1) M is foliated by umbilical (n - l)-submanifolds of E"+2; Pp, Pp as in the picture below: L (2) M is locally the envelope of a one parameter family (S"~ )( of (n — l)-spheres in Ht"+2 with radii T — r{t) = TOT an(l with centers on the focal points c = c(t) ~ f + r - £i with multiplicity n — 1 or re;
(3) At each point p, the focal set 7P can be represented by one of the following two figures:
Theorem 4 Let M be a conformally flat submanifold of codimension 2 in 1R2+" (n > 5) without umbiiic points, such that;
(ii) Vp G M, a,, i- 0;
(iii) fi = TTTI is qiiasiumbilical; Z • (iv) M possesses no umbilical direction. Suppose, moreover, that the normal Fia connection is flat. Then:
(1) M is foliated by umbilical (n-2}-submanifokls in 1R"+ .
Theorem 3 Let M £ CP£ (n> b) without umbiiic points, so that: (2) M is locally the envelope of a two parameters family (6""^2),,, of (n - -2)- (l)VpsM, (£,),,-T^M spheres in 1R™+2 with radii r = r(s) = ipr and with centers on the focal points c. -- c(«i 0 = / + r " £i °f multiplicity n — 1. (2) Vp e M, a,, / 0 (3) M admits an umbilical normal section. (3) At each point £ M, the focal set /,» can be pictured as below:
5 6 l. SOME RECALLS ON SUBMANIPOLDS [CH]
From now on, we consider an n-dimensional manifold M isometrically immersed in E™+". We denote similarly the canonical metric on JR.""*"" and the induced metric in M by <-,->. Let V and V be respectively the canonical connection of ]Rn+w and the v Riemannian connection in M. Put # (M) [resp. Jf-1 (Af)J the module of tangent [resp. ? t normal] vector fields. Denote by TM [resp. TXM] the tangent [resp. normal] bundle. 1.1 The second fundamental form (or the shape operator) is the bilinear symmetric map CT:*(M) x*(Af) — *±(M)
Theorem 5 Let Af be a 2-codimensional conformally flat submanifold in IR""1"2 For any [X, £) €^£ (M) XX1- (M), we decompose into a tangential part and a. normal (n > 5) without umbilic points, verifying the following three conditions: part: where (1) Vp eM, ap=fi 0; A [X)(E*(M), (ii) The normal direction a is not quasiumbilical; t Such a decomposition gives rise to two bilinear maps A and V-1-: (iii) The normal connection is flat. A:*(M) x
Then: (l) Vp £ M, (Ei)p = TfM. (2) M is foliated by umbilical (n-2)-submanifolds in JR™+2. V-1- :*(M) xx-1-(A/) ^^ (M)
(3) M is locally the intersection of a one-parameter family (E,)j of {"• - l)- 2 +2 spheres in IR"* with a two-parameters family (O,,t),,t of (n - 2)-spheres in 1R" . The Vx defines a connection on the normal bundle T^M, called the normal connec- spheres £s admit radii rj. = ri(s) and centers on foca! points cL = «i(s) with multiplicity tion; so it possesses a curvature tensor field: the normal curvature tensor field ^-L. wit n - 1 or n. The spheres 6,lt are centered on focal points c2 = c2(s>0 h multiplicity The normal connection is said to be flat if R1- vanishes identically. n - 1 or n; their radii are of the form r2 = r2[s,t), (For i = l,2,r; is equal to the distance from e; to /(p), for each p £ Af). The map A is the Weingarten tensor field. For any section £ of TXM, A deter- (4) The focal set J(p) over any point p g M is the union of the intersecting lines mines a symmetric endomorphism A$ :*(M) —»-X(M) (called the Weingarten tensor field P,,,Df,, A,, as shown as in the following picture: in the direction £); it is related to a by:
1.3 The mean curvature vector field of M in 1R"+JV is the normal vector field II — ~TT{IT), where Tr is the trace with respect to <•,•>. 1.4 The first principal normal space of the submanifold M is the subspace E\ of*-1- [M] spanned by the image of a. (see [MO])
1.5 We let £2 = {prEi(V^^)/(f,X) e Et x * (M}}, where prE± is the orthogonal 1 projection of X - (M) on the orthogonal supplementary ijj- of Et. The subspace E% of *•*• (M) spanned by £2 is the second principal normal space (see [MO]).
8 w 2. QUASIUMBILICAL SUBMANIFOLDS [CH] 3.1 The class SJn (compare with [ZA], Chap. 2) Let Af be a N-codimensional submanifold in 1R™+JV, 1 < N < n. 2.1 Let us begin by denning what is a quasiumbilical direction. An element £ of % (M) is quasiumbilical if there exist two functions Aj, JJ,^ a,nd 3.1.1 Definition a unit tangent vector field T such that: We say that M belongs to the class ST^, KM admits an open covering
At{X) = AjX + ^(
+W for any section X of TM. In this case, A( possesses an eigenvalue of multiplicity > n — 1. (1) for 0 < r < N, each Ur is foliated by umbilical submanifolds in IR™ of In particular: dimension n — N + r, x ± 1. if A^ = fit — 0, i.e. A$ = 0, £ is geodesic; (2) for each leaf £ the subbundle T M\v in T E is parallel.
2. £ is umbilical if fij = 0; 3.1.2 Lemma 3. £ is properly umbilical if fi£ = 0 and A^ / 0; Let M e STj? (1 < N < n), p e Af, Ethe leaf passing by p; put n-N+r = dimK. 4. if Aj = 0, f is cylindrical; aT1< m Then for any £ 6 X-^ (A^)|E J any orthonormal frame field {ei,...en} such that «w-r+ivien are tangent to H: 5. £ is properly cylindrical if A( = 0 and ft{ ^ 0; 6. if Aj ^ ^ij and A^ ^ 0, f is general quasiumbilical. A = 0
But S2 x S2 canonically immersed in E0 is not conformally flat. 3.1.4 Now, take M €- QO^, for 1 < JV < n.
There exist (locally):
3. PROOFS OF THEOREMS 1 AND 1-BIS - (1) JV unit tangent vector fields T\,---TN, FOCAL SETS AND QUASIUMBILICITY (2) JV mutually orthonormal normal vector fields £i,...,f.nr, and
Before defining the concept of focal points, we begin by introducing a class S7" (3) 2iV functions Xl,...,Xtf, ftL,...,Hn such that VAT, V r X (Af): containing the class Q8%, for 1 < N < n. N a{X,Y) - 52{A,, < X,Y > -r/x,, < -V,1',. >< y,T,, >}6. •
10 From this expression of the second fundamental form 3.2 The concept of focal point [MI] Property 2 The focal set (i.e. the set of all focat points) of (Af, p) is: Let / : M —* 1R"+JV be the isometric immersion of M in IR"+W (of codimenskm ^ = {/(P) + nh G r^Af and det^,, - /) = 0} N). where / is the identity of TPM. 3.2.1 Consider the endpoint map E : TLM -> 1R"+W which assigns to each pair (p, £) Property 3[ZA] If M £ S 7£, then for any p e Af: Jv = ?p U Qp, where of a point p€ M and a normal vector £ e T^M, the point E(p, f) = f(p) + £ e M"+N where mj is a (JV — r) x (JV — r) matrix determined in Lemma 3.1.2, and / is an identity matrix of order equal to the order of m;. In terms of Pp and Q,,, we are able to characterize a quasiumbilical direction and a quasiumbilic point (i.e. a point where the submanifold is quasiumbilical). 3.2.3 Lemma (Quasiumbilical direction and focal points) +jN Let p be a point on a submanifold M e SJj?, i a wnit vector in T-}-M, t^ the Roughly speaking, a focal point of Af is a point of IR" where nearby normals normal line by f[p) and directed by £. Then: intersect. (1) f is geodesic if and only if t^ n 7P = 0. (2) £ is cylindrical if and only if Pv n tj = 0 and Qp n t% is reduced to a focal point with multiplicity one. (3) £ is quasiumbilica) and noncylindrical if Pp n Qp n f{ is reduced to a focal point with multiplicity n — 1 or n. 3.2.4 Demonstration of Theorems 1 and 1-bis We remark that a quasiumbilical direction is either geodesic, or cylindrical or non More precisely, let p € Af; a focal point in (Af, p) is a point (2) 9 is a critical value of E. So we deduce easil- Theorems 1 and 1-bis from 3.2.2 and 3.2.3. 11 12 4. THE PICTURES OF FOCAL SETS OF A SUBMANIFOLD M 6 Qfl*, Situation dim [Ei)p - Pp*--- Qpi»... PpC\Qp\B... The picture of 7V = Pp U Qp n>2 1 0 empty- empty empty 2-1 1 empty a line Pp empty Now, we purpose to Bnd all possible forms and positions of Pp, Qp and 7P above Y a point p 6 M, where M 6 S7£, in order to get a quasiumbilic point (i.e. a point where ftp the submanifold M is quasiumbilkal). 2 2-2-1 2 empty the union of two Let M £ S/n (n > 2), p £ M. Then 7V = Pp U Qp, where: lines Dp, Ap (1) either Pp is empty, or a straight line, (2) Qp is a proper or improper conic. 2-2-2 2 empty a hyperhola empty When we Compute the equations of Pp and Qp, we obtain the Proposition which follows: Proposition 6 Assume that p is a point in M, where M £ $7%, n > 2. p is a quasiumbilic point if and only if one of the situations in the following table is satisfied: 3-1-1 1 a line the lineP,, 3-1-2 1 a line the union of PJ? with a line A?, where A,,///>, 3-2-1 2 a line with a line A,, 2 a line a line 3-2-3 2 a Line the union of d singltton { 2 lines Dv, Ap uch that Vj,IJPp. 13 1 1 1 ituation dim (Ui)p — PP n I?,, «. The picture of 7P = P,, U <2P Remark 1 If Af is either 5", or S x 1R"" , or S"" x IR, canonically immersed in +2 3-2-4 a Line a. hyperbola & singleton Ht" , then at any point p£ M, the focal set 7P is pictured in the situation 3-1-1. \ 1 1 Il+2 Remark 2 When we immerse, by a canonical way, S X S™" in IR ! we obtain K the situation 3-2-2. -fCf) * 3-2-5 parabola singleton 5. THE LOCAL GEOMETRY AND THE FOLIATED STRUCTURE OF A CONFORMALLY FLAT SUBMANIFOLD OF CODIMENSION N * 1 = 2 IN IR"+2 Kf) 3-3-1 a line the union of 2 lines the set of 2 points 5.1 Motivation Using the equality Q9^ = CP,\ for any n > 1, M. do Carmo, M, Dajczor and F. Mercuri studied in 1985 (|DDM]) the local geometry of a conformally flat hypcraurface Mt of ]R'l+1, which is neither cylindrical nor umbilical at any point. They stated that: a line 3-3-2 +I parallel lines Pr, 91,92 (1) Mi is foliated by umbilical {n - l)-submanifolds of Dl" , (2) it is locally the envelope of a one parameter family of (n - l)-spheies: [\f - e(()|| ~ T4TT = r(t), where a is the first canonical normal vector field of Mi. We attempt to generalize this statement in Snbsecs. 5.2, 5.3, 5.4, 5.5 and 5.6 by A line abola composed of means of the equality: 2 points <7i,lj2 C?l = Q$l, for n > 5 In addition, we will discuss on the first and the second principal normal spaces E^^E^ of the submanifold. 3-3-4 a line a hyperbola reduced to 2 pointa 9t,(?2 5.2 Proof of Theorem 2 Lemma 5.2.1 In accordance with the conditions of Theorem 2, M is general quasiumbilir.al in 3-3-5 line an ellipse reduced to the direction £i = jAt and geodesic in the direction £2 perpendicular to £j. 2 points (J 1,52 Proof This Lemma ie a direct consequence of the n° 5-1-2 and £-1-3, p. 64 of [ZA|. Now, we put X = | Q| and p =< a + nB,£i >. Lemma 5.2.2 The sets IS 16 Hn+2 being flat, we have: y 6 * (M), < ilt(X),y > = < a + rtS, { >< X,Y >} define two supplementary involutive and differentiable distributions on Af. Moreover, A is constant along each leaf of E\. Vp e M, dim(£i)p = n - 1, dim(E^)r = 1. (2-4-* *) 0 = Proof It is due to: (1) Lemma 7 and Theorem 2 of Chapter II in [ZA], for any (X, F, £) £ * (M) x -X (M) x *-•- {Af) (2) Theorem 1 and Proposition 2 of [RH], We apply this equation first to (V;,T, fa), and secondly to (V;,r, O) f°r * Now, we want to get local coordinate system, which are adapted to these folia- 2,... ,n. We obtain: tions. (i)-r(r) = o, u. Lemma 5.2.3 {2) Vv.T = -T^- Locally at each point of Af, there exists a local coordinate system (t,x2,...i") where A ^ fi {because of Lemma 5-2-1). This implies that: such that, if we put T = -jfe and V; = ^, for 2 < i < n: U) [Vi,Vj\ = 0 , for 2 < i < j < n, (2) [Vj,r] = 0,for 2 (2) £L is paralJel; and so is fo. As a Corollary of the Lemma 5-2-4 we have the Lemma 5-2 5. (3) [E2),, = 0, Vp e M. Lemma 5.2.5 4 Proof Put i{X) =< Vjcti,t2 >, \/X e * (Af). Proposition 8, p. 25 in |ZA] shows M can be considered as a conformally flat hypcrsurfacc of IR" '. that T(V;) = 0, for t = 2,..., n. Now we compute Proof This lemma comes from Lemma 1 in [MO], From the Weingarten equation of M and Lemma 5.2.3, we get the following system (2-4-*): Moreover, we can get now more precisions on the leaf of E\. Lemma 5.2.6 Let p € Af, J2), thc leaf of E>- Pass:ng by p, ft, x2,... ,x"-) the local coordinate system at p adapted to the .oliations B\ and /i'(i, according to the Lemma 5.2.3. Then: 17 18 1 2 (1) The leaf J2X is umbilical in M and in IR"" " . It is of constant sectional determine, on M, two differentiable integrable supplementary distributions such that, curvature; and its mean curvature vector field in TR.n+2 is parallel. VpeM : dim(£i)I, = n - 1, dim(£B)p = 1 . (2) The field T = £ is normal to £v Moreover, A is constant along each leaf of Ex- (3) Put e=/ + !$!, and r= f Proof The proof of this lemma uses Lemma 5.3.1 hereabove, and Theorem 2 in [ZA], Then c = c(t,i2,... , x") = c(t) Theorem 1 and Proposition 2 of [RHj. and r = r[t,x*,...zn) = r(i) Lemma 5.3.3 (4) Moreover, at p passes a (n-l)-sphere ^t of equation: At any point p e M, there exists a local coordinate system (t,x.2,x3,... ,1") such '(') = J = II/(P) - CMI1 "- ll/(t,*a. • • • ,*") - e(OII > ° • that: (l)[Vi,Vy]=0 Proof This Lemma is a corollary of: (1) the Lemma 5.2.4 (3) Au{VA=\Vi, Ai7(V,) = 0 (2) the Proposition 3.1 {p. 49), and the Proposition 3.2 (p. 50) of [CH) (4) Ail(T) = liiT, Au{T) = n,-T (3) the Lemma 5.2.2, for any i,j, 2 Lemma 5.3.1 (2) For 1 = 2,..., n, we have: According to all conditions of Theorem 3, M is umbilical in the direction £i = y^ = 0, (V ) = 0 and properly cylindrical in the direction £2 perpendicular to £1. Moreover, the normal n t connection of M in IR™"*"2 is flat. Proof This Lemma is a consequence of Lemma 2 (p. 80) in [ZA]. (3) VV,T = VrV; = A! ^ ^ Now, we put A = ||ex||, fii =< a + nB, & > for i = 1,2 Proof To prove this lemma: We get two foliations on M. (1) We appiy Proposition 8 (p.25) in [ZA]; Lemma 5,3.2 (2) We use the fact IR""*"2 is flat, and we write the Weingarten equation of /. The sets (3) We use Lemma 5.3.3. =< -X} x w = {X e * e * (M), =< 19 20 * i Lemma 5.3.5 The sets = {X € € ^ (Af), Let p€M,Yii. the leaf passing by p (for E>), ((, i2,..., zn) the local coordinate system at p adapted to the foliations E^,Ep. Then: (for j = 1,2) determine three differentiable involutive supplementary distributions such that, +2 (1) The leaf J^A is umbilical in 1R" . In particular, it is of positive constant VptM: Ai sectional curvature; its mean curvature vector field in E.""1"2 is parallel. Moreover, X is constant along each leaf of B*. (2) T = m is normal to EA- Lemma 5.4.3 (3) Put r = i and c = / + r^. There exists, at each point of M, a local coordinate system (s,t,x3,... , x") such c: an< a one We get two curves ^^(* ""'^^J^t) > i parameter family of (re - 1)- that: spheres of equations: (l)[Vi,Vy]=0 \\M-e[t)\\=r{t)>0 and passing by the point f(p). (3) [Vi,T]=0 Proof This lemma is deduced from: (1) Lemma 5.3.4. (5) Atl(Vi) - AV;, A(l[S) = ,HS, AU(T) = XT; (2) Proposition 3.1, p. 49 in [CH], (3) Proposition 3.2, p. 50 in |CH], where (4) Lemma 5.2.2. ol os ox* Lemma 5.3.6 lemma 5.4.4 The picture of the focal set is obtained from the situation 3.2.1 in Proposition 6 We put: T(X) =< V f!, £ >, (Ai). Then: of Sec. 4. x 2 5.4 Proof of Theorem 4 We use the same methods and the same plan as in Subsec, 5.3 to prove Theorem , Vs&> - 0 \. We get nearly the same lemmas. (3) For j = 3, ...,n: Lemma 5.4.1 a M is general quasiumbilica! in the direction Ci = jrjj] "d properly cylindrical in (4) S and r are normal to E\. the direction fj perpendicular to £i. Lemma 5.4.2 (6) Let By be the projection on £,- of 8, jiy —< a + nBj,Cj >, for J - 1,2, and A- Hat 21 22 Let (8) T(A) = 0 and A • (A - ^,) • M2 ^ 0 x E = {X £ * (M), A((X) =< a,Z > X} Lemma 5.4.5 ^ = {X e *^ (M) X£(X) -< a + nBl, f > -X} 1 tne ea e X (M), >l£(X) =< a + nB2, i > -X] We put r — j and c = f + r£L. Let p g M, Yix ' ^ °^ ^* passing by p, (s,t,i3,... ,i") the local coordinate system at p adapted to the foliations Ex^E^^E^. These three sets determine on M three supplementary differentiable involutive distribu- Then: tions such that Vp £ M: (1) The leaf £ is umbilical in M and in Hf1"1"2; its mean curvature vector field A ,, = n - 2, in K"+2 is parallel. (2) We get a curve r'^~^\ and a 2-parametcrs family of (n — 2)-sphores of Moreover the curvature function (£*( is constant covariant along each leaf of the foliation E, equations: Lemma 5,5.3 rl+2 and passing by the point /(p), where c = c(s,t) is a surface of lR . At each point of M, there exists a local coordinate system (s, t, i3,..., x'1) Remark 5.4.6 We obtain the picture of the focal set from the situations 3.2.2, 3.2.3, such that (1) \Vi,V3\ = 0 3.2.4, 3.2.5 of Proposition 6 in Sec. 4. 5.5 Proof of Theorem 5 Using the same plan of proof and the same methods as before, wo get the following (3) [Vi,r] = 0 successive lemmas. Lemma 5.5.1 (4) [S,T] = 0 There exists locally at each point an orthonormal frame field composed of general quasiumbilical normal directions. (5) ^(VO- Proof It is a consequence of p.66, n°/> - 2 in [ZA]. Now, we put for j = X,2: • Bj the projection of 8 in £j-; • fij =< a, + nBj, £j > • Aj =< o, £j- >. Then, from Lemma 5, we have: (6) We will choose £t and ^2 such that At > 0 and A2 > 0. where: V; = ^ and 23 • a two parameters family {flj,t)a,( of (n-2)-spheres of equation: ||/-c2(s,t)l| r2(si0 > 0, such that f(M) is locally on the intersection of these two families. =&. T = &, for2 Then: (1H(X)=O, (2) f (T) = 0, (3)V5e. = - Acknowledgments VSt2 = ->• (4) VVj£t = -AJVL, VVj£2 = -A2V; for 3 < i < n. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality a.t the International Centre for Theoretical (5)T(At)=0,r(7(5))=0. Physics, Trieste. (6) S and T are normal to E. (7) V^S = Al'1^L'^*(>'l) + x^f ' * = ^sVi' for 3 < i < JI (8} V^T = ~^t3} • vi + rf-*'J ' ^ = ^^ for 3 < i < n Lemma 5.5.5 (1) Each leaf of E is umbilical in M and in Dtn+1J; its mean curvature vector field in IR"+2 is parallel. (2) Put, for k = 1,2: 1 We obtain: ri anc • two curves at_^ u) ^ "',^(,1 • two surfaces r2j C2: r2 : ut —* ut c2 ! /i —^ ixt • a one parameter family (£„), of (n — l)-spheres of equations: 25 26 REFERENCES [CA] Cartan E., "La deformation des hypersurfaces dans I'espaces conforme reel a. n > 5 dimensions", Bull. Soc. Math. France, 45 (1917) 65-99. [CH] Chen B.Y., Geometry of Submanifolds, (Marcel DekUer 1973, New York). [CY] Chen B.Y, and Yano K., "Special quasi-umbilical kypersurfaces and locus of spheres", Atti. Ac. Naz. L'mcei, Serie VIII, Vol. LIII, Fasc. 3-4, 1972. [CV] Chen B.Y. and Verstraelen L., "A survey on quasi-umbilical submanifolds", Med. Van de Kon. Acad. Voor Wet. Lett. En. Sch. Kun. Van Belgic, Jaargang XXXVIII, 1976, n. 7. [DDM] Do Carmo M., Dajczer M. and Mercuri F., "Compact conformally flat kypersur- faces, Trans, of the Am. Math. Soc, Vol. 288, 1 (1985). [KN] Kuiper N.H., "On conformally flat spaces in the targe", Ann. Math. GO (1949) 916-924. [MI] Milnor J., Morse theory, (Princeton, New Jersey, 1963). [MM] Moore J.D. and Morvan J.M., uSons-variete's conformement plates de codimtn- sion quatre", C.R. Acad. Sci. Paris t. 287 (1978), serie A, 655-657. [MOj Morvan J.M. , "Cylindrieity", Riv. Mat. Univ. Parma (4) 8 (1982), 161-173. [MS] Spivak M., "A comprehensive introduction to differential geometry", (Publish or Perish Inc. Boston, 1975). [RH] Reckziegel H., "Completeness of curvature surfaces of an isometric immersion", J. Diff. Geom. 14 (1979) 7-20. [ZA] Zafindratafa G., "Sous-varicte's confomernent plates d'un espace eulidierF, These de 3e cycle, Universite de Provence - Marseille (1986)-France. 27 Stampato in proprio nella tipografia del Centre Internationale di Fisica Teorica