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					                                                    REFERENCE                                    IC/89/116
                                                                                      INTERNAL REPORT          1.          INTRODUCTION
                                                                                      (Limited Distribution)
                                                                                                                          Our purpose in this paper is to review all the already known results and iheopen problems
                                                                                                               concerning the conformal (lalness and the t|uasittmbil icily of submanifolds, in order to clarify the
',=X   1 2 JLM 1^39 \                 f International Atomic Energy Agency
                                                                                                               relation and the difference between these two topics.
                                                         and
                                    Ions Bducational Scientific and Cultural Organization                                A Riemannian manifold ( M n , < •, • >) is conform ally Hat if each point p G A/" belongs
                                                                                                               to an open neighbourhood U with local coordinates zl,..., x" such that:
                         INSTEttfjATIONAL CENTRETORTi 1E0RETICAL PHYSICS
                                                                                                                                                            ' ® dx]          dxn ® dx')

                                                                                                               where p € C°°(U) and n = dimiM"). It is well-known that if n = d\m(Mn) > 4, then
                                                                                                               (A/",< •, • >) is conformully flat if and only if its conformal Weyl curvature tensor field vanishes
                                                                                                               identically (for more information, see [CH11 Theor.5.1., p.26 or [GO]). Some very simple examples
                                                                                                               of such manifolds are the Euclidean space E ° , the hypersphere S" of E1**^ , a hyperbolic space
                                  A SURVEY ON CONFORMAL FLATNESS                                               H" and any space of constant sectional curvature. A non conformally flat manifold is for example
                              AND QUASIUMBIUCITY OF SUBMANIFOLDS *                                             the Riemannian product S2 x S1 of two 2-spheres.

                                                                                                                          We are especially interested in this work on conformally flat submanifolds.

                                          Georges Karoza Zalindratafa **
                                                                                                               2.          ORIGINAL PROBLEMS ON CONFORMAL FLATNESS
                             International Centre for Theoretical Physics, Trieste, Italy.
                                                                                                                          A W-codimensional submanifold M n of the Euclidean space E"*"w of dimension n+ N
                                                                                                               is called conformally flat if it is with respect to the induced metric from E * " . The study on such
                                                    ABSTRACT                                                   submanifolds began in 1917 with Elie Cartan:

                                                                                                               Factl (E. Cartan, 1917 |CAI] - J.A. Schouten. 1921 ISC]) The shape operator of a confonnally
                   Presently, some confusions ure made on the way of understanding the relation and Ihe        flat hypersurface of E ™+1, n > 5, admits an eigenvalue wiih order of multiplicity n — I or n.
         difference between the conformal flatness and Ihe quasiumbilicity of submanifolds. The purpose
                                                                                                                          Later on, N.H. Kuiper classified all compact simply connected conformally flat mani-
         of this paper is to make a clarification on these two lopics: We review all the up-to-date known
                                                                                                               folds.
         main results and the open related problems.
                                                                                                               Fact 2 (N.H. Kuiper, 1949 |KH|) Every simply connected compact conforniatly flat Riemannian
                                                                                                               manifold is globally conformal to a n-sphcrc S".

                                                                                                                          These two first results lead to three natural questions:
                                              M1RAMARE - TRIKSTE                                               Problem 1
                                                      June 19H9                                                1.1      Mow to generalize the Cartan-Schouteii's result (Fact 1) in codimension N > 2?
                                                                                                               1.2      Does it admit a converse in general?
                                                                                                               Problem 2 What is the global or local structure of a conformally flat compact subiiKinifolci?

                                                                                                               Problem 3 Is il possible to classify all compact confonnally flat submanifolds as did N.H. Kuiper
        * To be submitted for publication.
       *• Permanent address: University dc l'ianarynisou, Amlrainjaio, 301 l;iatiariintsoa, Madagascar.        in Fact 2?
          Since ihere are only very rure articles treating these last two problems (see for example      Fact 4 (B.Y. Chen and L. Verstraelen, 1976 |CV|) U t U be a W-codimensional submanifold
[DDM] or |MO1|) we will omit them. But it is very remarkable that many mathematicians, like              with a flat normal connection in the Euclidean space E ' " * ' , 1 < N < n— 3. If M is conformally
B.Y. Chen, M. do Carmo, M. Dajczer, R Mercuri, J.D. Moore, J.M. Morvan, L. Verstraelen,                  Hat, then it is quasiumbilical.
K. Yano, and some oihers tried or arc still trying to attack these three problems. Even long and hard
                                                                                                                   Unfortunately, Fact 4 ignores completely the case where:
is the way to obtain the required answers, a very big progress has been made, mainly for Problem
                                                                                                                   either the normal connection is non flat,
                                                                                                                   or the dimension is n = 3
                                                                                                                   or the codimension N and the dimension rcarc related by N > n— 2 > 2.
                                                                                                         This situation motivated J.D. Moore and J.M. Morvan to consider only small codimensional sub-
3.         AN EXTRINSIC CIIARACTKRIZATION OF CONFORM ALLY FLAT                                           manifolds but without any hypothesis on the normal connection.
          SUBMANIFOLDS: THE O.UAS1UMBILICITY                                                             Fact 5 (J.D. Moore and J.M. Morvan, 1978 [MM]) Let M be a submanifold of codimension JV in
                                                                                                         E'H^.with j < w <- inf(4,n— 3). Then M is conformally flat if and onlyif it is quasiumbilical.
           In order to resolve Problem 1, B.Y. Chen introduced in 1972 ICY] quasiumbilical sub-
manifolds; the second fundamental form of such suhinanifolds is nothing but a direct generalization                  From these last two results, we reformulate our main Problem 1 to get the following one.
of that one of a conformally flat hypersurface us Cartan and Schouten found.                             Problem 4
                       n                                                               +w
           L e t / : M *-* E**" be an isometric immersion of codimension JV in E " . A normal            4.1      Does there exist any conformally flat submanifold M of codimension N in E**" with
vector field £ of M (or / ) is quusiumbilical if its shape operator A( can be represented by a matrix:   W > n — 2 o r A f > 5 , which is not quasiumbilical'.'
                                                                                                         4.2     Does the equivalence 3.1 (in Fact 3 hereabove) hold true for a 3-dimensional submanifold
                                                          0
                                          0    V                                                         4.3     Is it possible to find a quasiumbilical isometric immersion with a non flat normal connection?

                                                                                                                  Already in 1973, G.M. Lancaster answered negatively to Problem 4.2 in the case of codi-
                                       \ 0     ...   0        \t)
                                                                                                         mension N = 1 by an existence theorem.
where Xj and fi( are some functions depending on £. The submanifold M (or the isometric immer-           Facl 6 (G.M. Lancaster, 1973 [LA]) There exist conformally flat hypersurfaces of E 4 which are
sion / ) is (totally) quasiumbilical if, locally around each point of M, there exists an orthonormal     not quasiumbilical.
system composed with N normal quasiumbilical vector fields.
                                                                                                                  But 54 years before, E. Cartan resolved in his article published in the "Bulletin de la
          With this new concept B.Y. Chen and K. Yano reformulated Fact 1 hereabove and im-              Societe Mathematique de France" the same problem 4.2 for the flat case and for any codimension
proved it as follows:                                                                                    N>\.
Fact 3 (B.Y. Chen and K. Yano, 1972 |CY] or |CIII ])

3.1 A hypersurface of E M', n > 4 is conformally flat if and only if it is quasiumbilical.               4.           CYLINDRICITY AND FLATNESS OF SUBMANIFOLDS
3.2 Every quasiumbilical submanifold of codimension N in the Euclidean space E " N, n > 4,
is conformally flat.                                                                                                 The second fundamental form a of a Hat submanifold M of codimension N in
                                                                                                         satisfies the Gauss equation:
          Actually, B.Y. Chen proved in 1974 |CH2| that the multiplicity of the principal curvatures
of a submanifold of a Riemannian manifold is conformally invariant. Hence Fact 3 is also true if                                     < a ( x , j/), a(2,ui) > - < o(x,z),            a{y,w)     > = 0
we replace the Euclidean ambient space by any confonnally flat space having the same dimension.
                                                                                                         for all tangent vector fields x}y,z, w of M , where < •, • > denotes the canonical metric of I;"' N.
Moreover, it is easy to prove that the statement 3.2 holds still true in dimension n > 3.
                                                                                                         Any bilinear symmetric mapping h : E " x H — + HN satisfying such equality
                                                                                                                                                                       ™
            In 1976, B.Y. Chen collaborated wilh L. \trstraeten and obtained the lirst powerful gen-     (i.e. < h ( x , y ) , h ( z , w ) > — < h ( x , z ) , h ( y , u > ) > = 0 for all x , y , z , w € E " w h e r e < •, • > is
 eralization of the Cartan Schotiten's ilicorcm.                                                         the scalar product in E N) is called flat.
            Before slating Cartan's solution for the problem 4.2, let us recall some surprising lemmas   Problem 5
on flat bilinear symmetric rmips (for (lie proofs or more information, see E. Cartan, 1919 1CA2| or      5.1    Determine all flat bilinear symmetric maps fc : E 4 x E * —>E N where N = 5 or 6 .
O.K. Zafindratafa and J.M. Morvan, 19K6 |MZ1 or J.D. Moore, 1972 [MO2J).                                 5.2    Determine ail flat bilinear symmetric maps t : E 4 x E 4 — E w , /V = 5 or 6, which are
                                                                                                                                                                           >
Lemma 7 Let h : E " x E ° — E w be a Hat symmetric bilinear map. lf£ e E N is a cylindrical
                                   »                                                                     not cylindrical (or not quasiumbilical).
direction for h, then the projection of k on the hypersurface £ x of E N onhogonal to £ is also a flat
bilinear symmetric map.
                                                                                                         S.        NON QUASIUMHILICAL, CONFORMALLY FLAT SUBMANIFOLDS
         We recall that £ € E N is a cylindrical direction for h if the quadratic form < h,( >
admits 0 as an eigenvalue with multiplicity n— I ornon E".
                                                                                                                   Untit now, a general method of resolution of Problem 5 is still unknown. However, J.M.
Lemma 8 L e l / i : E " x E " - i C be a flat bilinear symmetric map. If Ker(h) = {()}, then             Morvan and the author gave, in 1986, a punctual solution to Problem 5.2 for the case when N = 6.
h is cylindrical.
                                                                                                         Facl 11 (J.M. Morvan and G. Zafindratafa, 1986 [MZ]) Consider the 4-submanifold M 4 of E l 0
                                                                         w
         We recall that a bilinear symmetric map /i : E " x E " — E is cylindrical if there exists
                                                                 »                                       given by the parametrization:
an orthonormal frame in E N composed with N cylindrical directions.
Lemma 9 Let h : E " x E n — E N be a flat bili near symmetric map, where 1 < n < 3. Then
                           >
h is cylindrical.                                                                                        where
           This last lemma implies obviously the first part of Fact 10 which follows:                                                Mx,y,z,t)      = - < i 2 + y2 + z1 + t2) + zt

Fact JO (E. Cartan, 1919 [CA2| - or J.M. Morvan and G. Zafindratafa, 1986 [MZ])                                                              ,z,i) - xy
                                                                                                                                     f3(x,y,z,t) = yz
10.1 Every n-dimensional flat submanifold of E n+W with 1 < n < 3 is cylindrical. Moreover,
E. Cartan stated without proof that:
10.2 There exists a JV-codimensionalflatsubmanifold of E " + w , n > 4, which is not cylindrical.                                    Mx,y,z,t) = x(z + t)
           In order to justify this free statement 10.2 we analyze the particular case of dimension                                  / s (z, !/,*,« = j(z2 + tz)
n=4.
                                                                                                                                     fi(x,y,*,t)    = -x + y+ z + t
           Assume h : E 4 x E 4 -+ E w is a flat bilinear symmetric map, and consider the dimen-                                     fi(.x,y,z,t)   = x-y   +z + t
sion of the space | Im( h) ] generated by the image of h. We can suppose, without loss of generality,
that N - d\m\ /m h\. Since the dimension of the space of all symmetric bilinear forms on E 4 is                                      fa{x,y,z,t)    = x + y - i +t
equal to 10, we can restrici ourself lo: 0 < N < 10.                                                                                f\aix,y,z,t)    <*x + t/+z — t

        Using technics similar to those in the proof of Lemma 9 it is easy to demonstrate that, if       The Gauss equation shows that M* is flat at llie particular point 0. But M4 admiis no t]uasium-
 N e {7,8,9,10}, there exists in | /m(h)\ at least one cylindrical direction. Hence by Lemma             bilical normal direction at 0; hence 0 is neilher a cylindrical point nor a quasiumbilical point for
 7 we may reduce N and turn up to ihecaxe when N = Uiin| Im(li) \ £ {(), 1 , 2 , 3 , 4 , 5 , 6 } . But   M4.
 Lemmas 7,8 and 9 imply together thai, if N e {(), 1,2, 3,4 }, ihe Hat bilinear imip h is cylindrical.             In addition, during the tirst year stay of the author at the International Centre for Theo-
 Consequently, there are only two unknown .situations: "JV = 5"a»d"JV = 6". We summarize this            retical Physics in Trieste, he gave a punctual answer to Problem 4.1 for the particular case N =
 discussion in the following linear algebra problem:                                                     n - 2 = 2.
FactI2(G.K.Zafindraiafa, 19K8[ZA2|) L e t M 4 be the open subset of E« given by the parameiriz.a-        for all tangent vector fields X, T where T is a unit tangeni veclor Held, A and 8 are a normal vector
t i o n / = ( / i , . . . , / f t ) : h.A -> E * in l i " which follows:                                 fields on M,


                       fdx,y,z,t) = |                                                                                               ai.x,Y)     =A- <X,Y                                                               (2)


                                                                                                         for all tangent vector fields X,Y where {T\,T2} isanorthonormal system of tangent vector fields,
                       h(x,y,z,t)        = x                                                             .4 is a normal vector field, and B], B 2 form an orthonormal frame field on the normal space. More-
                       f4(x,y,z,t) = y                                                                   over if the case (1) occurs then every normal direction of ( M , / ) is quasiumbilical.

                       fs(x,y,z,t) ~ z                                                                   Fact 15 (G.K. Zafindratafa, 1986 [ZA1]) Lei </>, : E " ~> E n + I be a cylindrical immersion with
                       f,,(x,y,z,t) = t                                                                  a second fundamental formCTIgiven by:

where e = ± 1.                                                                                                                                ffiCX.n =<X,Tt ><Y,T, >^i

         The conformal Weyl curvature tensor of the 4-submanifold (M*, / ) vanishes at the point         for all tangent vector fields X,Y where Tj is a unit tangent veclor field and £\ is a unit normal
0 . Moreover the normal curvature tensor of M 4 i n E f i vanishes (i.e. the normal connection is Hat)   veclor field of <^].
at 0 . Nevertheless (M*,/)   admits no quasiumbilicat normal direction at 0 ; hence 0 is not a
                                                                                                                  lxl 02 : E n + I — E " * 2 be another cylindrical immersion with a second fundamental
                                                                                                                                    >
quasiumbilical point.
                                                                                                         form oi such that:
         Independently and without knowing this example of Fact 12, U. Lumiste and M. Viiljas                                                 <72(X,Y) =< X,T2 >< Y,T2 > 6
showed in 1989 the existence theorem he low.
                                                                                                         for all langeni veclor fields X, Y of 4>2 where T7 is a unil tangeni vector field non perpendicular to
Fact 13 ( 0 . Lumiste and M. Valjas, 1989 (LVJ) There exist JV^codimensional conformally flat            Ti and £2 is a unit normal vector field orthogonal to £,.
submanifolds with flat normal connection in E w W , N > n - 2 > 2 , which are not totally
                                                                                                                   Then / = ^2 0^1 : E n — E w 2 is a quasiumbilical (since cylindrical) immersion with
                                                                                                                                               >
quasiumbilical.
                                                                                                         non flat normal connection in E 1 * 2 ; its second fundamental form a can be expressed as follows:
            Now we want to discuss the last part (no. 4.3) in Problem 4 in order to show thai the
                                                                                                                             a(X,Y) =<X,T\ >                                         ><Y,T2
converse of Fact 4 due to B. Y. Chen and L, Verstraelen may be false.
                                                                                                         for all vector fields X, Y tangent lo / .

                                                                                                                   Independently, U. Lumiste and M. Viiljas improved recently the necessary and sufficient
6.          QUASIUMBILICAI. SURMANIFOLDS WITH NON FLAT NORMAL                                            condition in Fact 14.
            CONNECTION
                                                                                                         Fact 16 ( 0 . Lumiste and M Valjas, 1989 [LV]) A totally quasiumbilical submanfold M n of codi-
                                                                                                         mension N in I i w W , n > 4 , has flat normal connection if and only if there exists an orthononnal
           In his thesis of Doctorat de 7>' Cycle in 1986, the author presented an example of a cylin-
                                                                                                         tangent frame field T\,..,, Tn (of M) such that:
drical (hence quasiumbilical) 2-codimensional submnnifold in E n t I with non flat normal connec-
tion. Moreover he showed a necessary and sufficient condition (on the second fundamental form)           (1) {T*i,... ,Tn} diagonalizes simultaneously all Weingarien operators of M in F,n*N,
for a 2-codimensional (juusiumbilical submanilold of li"^ 2 to be with flat normal connection.           (2) for any orthonomiai system {£1,... ,£w} of quasiumbilical normal directions, there exist 2 N
                                                                           n   2                         functions X l , . . . , > w l ^ 1 , , . . l / j A / such the Weingarten operator A(a for each £ a can be expressed:
F a d 14 (G.K. Zalindraiafa, 19K6 |ZA1|) Let / : M -> V, * be a quasiumbilicat isometric
immersion of a Ricmannian n-munifolJ ( M, < •, • > ) in I i n t 2 . The normal connection of / is                        < AU(X),Y         > = \"<X,Y           >+,i« <X,Ttln)         > •< Y,Tkia) >
Hat if and only if ils second fundamental form a can lie written as follows:
                                                                                                         for all langent vector fields X, Y where k(a) C {I               n} depends on a, and < •, • > is the metric
                       cither       a(X,Y)     = A- < X,Y > + < X,T > < Y,T > B                  (1)     induced by li"*N on M'.

                                                             7
                                                                                                                                            REFERENCES
                                       Acknowledgments


           The aulhor would like lo express his sincere gratitude to Professor l^copold Verstniclen
                                                                                                      |CA1] Cartan E., "La deformation des hypersurfaces, dans l'espace conforme reel a n > 5
for inviting him for two weeks at Ihe Department of Mathematics, Catholic University of Leu-
                                                                                                            dimensions", Bull. Soc. Math. France 45 (1917) 65-99.
ven, Belgium, where this work was done. He would like to thank all Ihe Mathematicians of this
University and also L. Verstraclen's family for their kind hospitality. He would also like to thank   ICA2] Cartan E., "Sur les varietes de courbure constante d'un espace eucUdien ou non-euclidien",
Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at                 Bull. Soc. Math. France 47 (1919) 125-160.
the International Centre for Theoretical Physics, Trieste.
                                                                                                      ICH11 Chen B.Y., Geometry ofSubmanifolds (New York, 1973) M. Dekker.
                                                                                                      [CH2] Chen B.Y., "Some conformal invariants of submanifolds and their applications", Boll.
                                                                                                            Un. Mat. It. (4) 10 (1974) 38O-385.

                                                                                                        [CV] Chen B.Y. and Verstraelen L., "A characterization of totally quasiumbilica! submanifolds
                                                                                                             and its application". Boll. Un. Mat. It. 5,14A (1977) 49-57; Errat. corrige: ibid. (1977)
                                                                                                             Vol.A14, 634.

                                                                                                        |CY| Chen B.Y. and Yano K., "Sous-varietes locnlement conformes a un espace euclidien",
                                                                                                             C.R. Acad. Sci. Paris, serie A, 273 (1972).
                                                                                                      [DDMJ Do Carmo M., Dayczer M. and Mercuri R, "Compact conformally flat hypersurfaces",
                                                                                                            Trans. Amer. Math. Soc. 288 (1985) 1.
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                                                                                                        [LA] Lancaster G.M., "Canonical metrics for certain conformally Euclidean spaces of dimen-
                                                                                                             sion three and codimension one". Duke Math. J. 40 (1973) 1-8.
                                                                                                        [LV] Lumiste t). and Viiljas M., "On geometry of totally quasiumbilical submanifolds", Tapt.
                                                                                                             YH.TA, 836 (1989) 172-185.

                                                                                                       [MM| Moore J.D. and Morvan J.M., "Sous-varietcs conformement plates de codimension qua-
                                                                                                            tre", C.R. Acad. Sci. Paris, serie A. 287 (1978) 655-^57.
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                                                                                                            (1977) 85-97.
                                                                                                      |MO2] Moore J.D., "Isometric immersions of space forms in space forms", Pacific J. of Math.
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                                                                                                        [MZ1 Morvan J.M. and Zalindnuafu G.K., "Conformally flat submanifolds", Ann. Fac. Sci.
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                                                                                                                                                    H)
       [SCI Schoulen J. A.."Ubertlic Konfonne AbbildungTuliinensioiialerMarinigfultigkdlen mit
            ijuadnnischer inassbcsliminung ;iuf eine Mannigfaltigkeit mit euklidischcr massbeslirn-
            mung", Malh. Z. II (1921) 58-88.

      |ZA1| Zaiindratafa G.K., Sotis-varieles coiiformcnicnl plales d'un espace euclidicn. These de
            doctoraldu 3* Cycle, Universite de Provence - Marseille (Faculle des Sciences d'Aviyrion),
            France, 1986.

      |ZA2] Zafindratafa G.K., "Remarques sur Ies sous-varie'te's a connexion normale plate, satis-
            faisunt la 7J • C-condtion ou la C •TC-condtion",ICTP, Trieste, Preprim N0.IC/H8/112
            (1988).




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