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Reference Ic/88/392 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 <?7, for any ^, study the normal bundle and try to get a simple expression of the second fundamental a < 7 < N. form by means of the Gauss equation. It leads to a subclass of CP^: the class QO^ of Theorem 1-bis Suppose we can find a point p on a submanifold M £ S7^ (1 < all jV-codimensional quasiumbilical submanifolds of ]R"+W (introduced by B.Y. Chen in N < n), where the first canonical normal vector field a does not vanish (i.e. P -f- 0). 1972 [CY]). The shape operator of an element of such a class generalizes directly the shape v operator of a hypersurface belonging to CP^: there exists, at each point, an orthonormal Then p is a quasiumbilic point if and only if there is an affine orthogonal iV-frauie frame {£i,... £;/} of the normal space such that, for each £„, \ < a <N, the Wcingarten +N {/(p);«l.---.9«.?a+l.-...9/J,9/1+1,.-.,1/JV} in Ht" satisfying: tensor A(a admits an eigenvalue with multiplicity n - 1 orn. (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.
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