Pseudo-spherical and pseudo-hyperbolic ...

Arch. Math. 68 (1997) 520 – 528 0003-889X/97/060520-09 $ 3.30/0 © Birkhäuser Verlag, Basel, 1997

Archiv der Mathematik

Pseudo-spherical and pseudo-hyperbolic submanifolds via the quadric representation, I By ANGEL FERRA´NDEZ, PASCUAL LUCAS and MIGUEL ANGEL MERON˜O *)

Abstract. We consider the quadric representation of a submanifold into non flat pseudo-Riemannian space forms. Then we classify submanifolds such that the mean curvature vector field of its quadric representation is proper for the Laplacian. We have got a complete characterization of hypersurfaces whose quadric representation satisfies ~H ~ ˆ lH ~ ‡ m…f ÿ f0 †. As for surfaces into De Sitter and anti De Sitter worlds we have D also found nice characterizations for minimal B-scrolls and complex circles.

1. Introduction. The quadric representation of a submanifold has become a very useful tool in certain classification problems of Riemannian submanifolds (see [3], [4], [5], [9] and [16]). In [13], in order to classify constant mean curvature surfaces into non-flat pseudoRiemannian space forms, we brought the quadric representation into the realm of indefinite space forms. Recently, in [12] and [14], the quadric representation has been also defined for product of pseudo-Riemannian submanifolds and has been used to solve some open questions related with a Chen's conjecture on biharmonicity (see for instance [8]). The purpose of this paper is to present the quadric representation in the most general setting and then look for a classification of submanifolds such that the mean curvature vector field of its quadric representation is proper for the Laplacian. The reason for dealing with this condition is twofold. On one hand, viewing the quadric representation as an isometric immersion, we pointed out that is a natural assumption in ~H ~ ˆ lH ~ has provided a terms of finite type submanifolds. On the other hand, the equation D source of properties for indefinite submanifolds without counterparts for Riemannian ~H ~ ˆ lH ~ allows submanifolds (see for instance [2], [10], [11]). Furthermore, we know that D to get 1-type and null 2-type submanifolds, as well as infinite-type submanifolds (see [7]). Here, in Part I, we provide preliminary computations and examples which we need in Part II. Since Parts I and II represent a whole, we also give in advance the main results of our work which are contained in Part II. Our main theorem gives a complete characterization of ~H ~ ˆ lH ~ ‡ m…f ÿ f 0 †. hypersurfaces whose quadric representation satisfies the equation D As a consequence, in dealing with surfaces into de Sitter and anti de Sitter worlds, we have Mathematics Subject Classification (1991): 53C50. *) This paper has been partially supported by DGICYT grant PB94-0750-C02-02 and Consejería de Cultura y Educacio´n de la Comunidad Auto´noma de la Regio´n de Murcia Programa Se´neca COM-05/96 MAT.

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got nice characterizations, among others, for minimal B-scrolls and complex circles. A ~H ~ ˆ lH ~ is also characterization of submanifolds whose quadric representation satisfies D given. ‡2 be the pseudo-Euclidean space of dimension m ‡ 2 with metric h; i having 2. Basics. Let Rm s a matrix, with respect to the standard coordinate system, given by G ˆ diag‰e1 ; . . . ; em‡2 Š, where e1 ˆ . . . ˆ es ˆ ÿ1 and es‡1 ˆ . . . ˆ em‡2 ˆ 1. Let us denote by SA…m ‡ 2; s† the space of selfadjoint operators on Rsm‡2 , that is, SA…m ‡ 2; s† ˆ fP 2 gl…m ‡ 2; R† : Pt G ˆ GPt g, Pt standing for the transpose of P. Let Ssm‡1 …r† and Hsmÿ‡11 …ÿr†, r > 0, be the central ‡2 defined by hyperquadrics of Rm s Ssm‡1 …r† ˆ fx 2 Rsm‡2 : hx; xi ˆ r2 g;

Hsmÿ‡11 …ÿr† ˆ fx 2 Rsm‡2 : hx; xi ˆ ÿr2 g: These hypersurfaces are of constant curvature 1=r2 and ÿ1=r2 , respectively. Without loss of generality, assume that r2 ˆ 1. For short we will write M m‡1 …k†, k ˆ 61, to indicate Ssm‡1 …1† or Hsmÿ‡11 …ÿ1†, according to k ˆ 1 or k ˆ ÿ1, respectively. Let us consider the map f : M m‡1 …k† ! SA…m ‡ 2; s† defined by f …u† ˆ uut G, where u is regarded as a 1-column matrix. It is easy to see that f is an isometric immersion provided k that SA…m ‡ 2; s† is endowed, as usual, with the metric ~g…P; Q† ˆ tr…PQ†. Then f is said to 2 be the second standard immersion of M m‡1 …k† into SA…m ‡ 2; s†. This map has been deeply studied in the Riemannian case (see for instance [5], [9] and [16]).  m‡1 …k†, the normal space of M m‡1 …k† in SA…m ‡ 2; s†, at f …u†, is At any point u 2 M Tf?…u† M m‡1 …k† ˆ fP 2 SA…m ‡ 2; s† : …P ÿ lI †u ˆ 0; Rm for some l 2 Rg;

~ of f I being the identity matrix. Thus f …u† 2 Tf?…u† M m‡1 …k†. The second fundamental form s is given by

…I 1† :

~ …X ; Y † ˆ …XY t ‡ YX t †G ÿ 2k hX ; Y i f …u†; s

for X, Y in Tu M m‡1 …k†. It is well known that M m‡1 …k† is minimally immersed by f k in a pseudosphere or a pseudohyperbolic space of SA…m ‡ 2; s† centered at I (see m‡2 [13]). Let x : Mnn ! M m‡1 …k†  Rsm‡2 be an isometric immersion and consider the new isometric  …Mn † be immersion f ˆ f  x, that will be called the quadric representation of …Mnn ; x†. Let X n the c 1 …M†-module of smooth vector fields on x.  …M n † 2 X  …Mn † ! c 1 …Mn ; SA…m ‡ 2; s†† by Define a map F : X n n n F…X ; Y † ˆ …XY t ‡ YX t †G; where c 1 …Mnn ; SA…m ‡ 2; s†† is the algebra of differentiable functions from Mnn into SA…m ‡ 2; s†. Then it is not difficult to see that F is symmetric, c 1 …M†-bilinear and parallel, that is, ~ rZ …F…X ; Y †† ˆ F…rZ X ; Y † ‡ F…X ; rZ Y †, r~ and r standing for the pseudo-Riemannian connections on SA…m ‡ 2; s† and Rsm‡2 , respectively. The endomorphism F…X ; Y † is characterized by F…X ; Y †…Z† ˆ hY; ZiX ‡ hX ; ZiY.

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Notice that, in the Riemannian case, this formula has a simple geometric meaning: if X and Y is an orthonormal basis for a plane P, then F…X ; Y † is zero on P ? , and on P is a symmetry sending X to Y and Y to X.  and H ~ denote the mean curvature vector fields associated to Let H  m‡1 …k†  Rm‡2 and f, respectively. Then an easy computation shows that x : Mnn ! M s

…I 2†

ˆ F…x H† ‡ n1 trF Pn where tr…F† ˆ ei F…Ei Ei †, fEi gniˆ1 being a local orthonormal frame tangent to Mnn and iˆ1 ei ˆ hEi Ei i.  …Mn †, define a map Y Z : X  …M n † 2 X  …Mn † ! c 1 …Mn ; SA…m ‡ 2 s†† by Given Z 2 X n n n n Y Z …X Y † ˆ F…X rY Z†. ~ will denote the Laplacians associated to r and r ~ , respectively. In what follows D and D ~ H

:

;

;

;

;

;

;

;

Then we have the following Lemma 1. With the above notations we have

…I 3† :

~H ~ D

ˆ ÿnF…H H† ‡ F…x DH† ‡ n1 D~ …trF† ÿ 2tr…Y H † ;

;

:

P r o o f . Let p 2 Mnn and consider a local orthonormal frame fEi gniˆ1 such that rEi Ej …p† ˆ 0, r being the pseudo-Riemannian connection on Mnn . Then

~H ~ …p† ˆ ÿ D

n X iˆ1

~ …p†: ~ Ei r ~ Ei H ei r

A straightforward computation yields r~ E r~ E H~ …p† ˆ F…rE Ei ; H†…p† ‡ F…x; rE rE H†…p† i

i

i

i

i

‡ 2F…Ei ; rEi H†…p† ‡ n1 r~ Ei r~ Ei …trF†…p†; and then the lemma follows. ~ …trF† by applying this Now we are going to give a complete description of D m‡1 …k††. Let p 2 Mnn and fEi gniˆ1 be endomorphism on Z 2 X…Mnn † and x 2 X? …M nn †  X…M a local orthonormal frame tangent to Mnn such that rEi Ej …p† ˆ 0. Then ~ …trF†…p† ˆ 2 D

n X

and so

iˆ1

~ …trF†…Z†…p† ˆ 2 D

ei F…DEi ; Ei †…p† ÿ 2

X

ei ej F…rEj Ei ; rEj Ei †…p†;

i ;j n X iˆ1

ÿ4

ei fhDEi ; ZiEi ‡ hZ; Ei iDEi g…p†

X

ei ej hrEj Ei ; Zi…p†rEj …p† Ei :

i ;j

By using the Gauss and Weingarten formulae we find that

rE rE W …p† ˆ j

j

n X iˆ1

ei fhrEj rEj W ; Ei iEi ‡ 2hrEj W ; rEj Ei iEi

‡ 2hrE W Ei irE Ei ‡ 2hW rE Ei irE Ei ‡ hW rE rE Ei iEi ‡ hW Ei irE rE Ei g…p† ;

j

;

j

;

j

j

;

j

j

j

j

;

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and then DW …p† ˆ …DW †T …p† ÿ 2

ÿ2

X

X

ei ej hrEj W ; rEj Ei iEi …p†

i ;j

ei ej hrEj W ; Ei irEj Ei …p† ÿ 2

X

i ;j

‡

n X iˆ1

ei ej hW ; rEj Ei irEj Ei …p†

i ;j

ei fhW ; DEi iEi ‡ hW ; Ei iDEi g…p†:

At the point p, this equation yields

~ …trF†…Ek † ˆ 4 D

X

ei ej fhrEj Ek ; rEj Ei iEi ‡ hrEj Ek ; Ei irEj Ei g

i ;j

‡ 2…DEk †? X ˆ 2…DEk †? ‡ 4 ei ej hs…Ej

;

Ek †; s …Ej ; Ei †iEi ;

i ;j

s being the second fundamental form of Mnn in Rsm‡2 . On the other hand, it is easy to see that

…DEk †? …p† ˆ ÿnr?E …p† H k

;

r? standing for the normal connection of Mnn in Rsm‡2 . Write s…Ej Ek † ˆ P dr hSr Ej Ek ixr , rˆ1 where fx1 . . . xmÿn‡2 g, xmÿn‡2 ˆ x, is a local orthonormal frame of normal vectors to Mnn in Rsm‡2 , dr ˆ hxr xr i and Sr denotes the shape operator associated to xr . Then mÿn‡2

;

;

;

;

X

X

mÿn‡2

ei ej hs…Ej ; Ek †; s …Ej ; Ei †iEi …p† ˆ

r ˆ1

i ;j

X

mÿn‡1

ˆ

r ˆ1

!

dr S2r

…Ek †…p†

dr S2r ‡ kI

Therefore

X

mÿn‡1

~ …trF†…Ek †…p† ˆ ÿ2nr? H ‡ 4 D Ek …p†

rˆ1

so that

…I 4† :

~ …trF†…Z† ˆ ÿ2nr? H ‡ 4 D Z

for all Z 2 X…Mnn †. Similar computations lead to

…I 5† :

;

X

mÿn‡1 rˆ1

!

…Ek †…p†

:

!

dr S2r

‡ kI …Ek †…p†

;

!

dr S2r

‡ kI …Z†

;

~ …trF†…x† ˆ 4nH D

and

…I 6† :

~ …trF†…x† ˆ ÿ2ntr…Y †…x† ÿ 4 D H

X

mÿn‡1 r ˆ1

dr tr…Sx  Sr †xr ‡ 4k tr…Sx †x;

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A. FERRA´NDEZ, P. LUCAS and M. A. MERON˜O m‡1

for any x 2 X? …Mnn †  X…M

…I 7† :

…I 8† :

…k††. From equations (I.3) – (I.6) we have

~H ~ …Z † ˆ 4 S ‡ 1 D H n

X

mÿn‡2 rˆ1

!

dr S2r

~H ~ …x† ˆ ÿ 4tr…Y H †…x† ÿ 4 D n

‡

ARCH. MATH.



X

…Z† ÿ 4r?Z H ‡ hDH Zix

mÿn‡1 r ˆ1

;

;

dr tr…Sx Sr †xr ÿ 2nhH ; xiH



4 2knhH ; xi ‡ hDH; xi ‡ ktr…Sx † x; n

‡ 2…n ‡ 2†H ‡ 2…hDH xi ÿ k…n ‡ 2††x ‡ k…DH†? H being the mean curvature vector field of Mnn in M m‡1 …k† and …DH†? the component of DH normal to Mnn in M m‡1 …k†. Note that equations (I.7) – (I.9) completely characterize the

…I 9† :

~H ~ …x† ˆ k…DH†T D

;

;

~ H. ~ endomorphism D ~ …Z† ˆ …2=n†Z, H ~ …x† ˆ hH ; xix and H ~ …x† ˆ kH ÿ x. It is also quite easy to see that H Finally we state the following useful result, just obtained in the Riemannian case in [9].

Lemma 2. Let fE1 ; . . . ; Em‡1 g be a local pseudo-orthonormal frame tangent to M m‡1 …k†. Then at every point x 2 M m‡1 …k† we have I

ˆ kf …x† ‡

X

m‡1 iˆ1

ei Ei Eti G:

3. A 2-type equation for hypersurfaces: Examples. Throughout this section we shall deal with hypersurfaces Mnm in M m‡1 …k†. Let N be a unit normal vector field to Mnm in M m‡1 …k†, S the shape operator associated to N and a the mean curvature function of Mnm in M m‡1 …k† etr…S† defined by a ˆ . The following formula for DH can be found in [6, Lemma 3], m

…I 10† :

DH ˆ 2S…ra† ‡ emara ‡ …Da ‡ eatr…S2 † ‡ kma†N ÿ km…k ‡ ea2 †x;

where e ˆ hN ; N i. Assume now that the quadric representation f satisfies the equation

…I 11† :

~H ~ D

ˆ lH~ ‡ m…f ÿ f0 †

for some real constants l and m, f0 being a constant matrix. It is not difficult to see that a hypersurface of finite type less than or equal to two satisfies (I.11). However, the converse does not hold as we have pointed out in [1]. For convenience, we shall call A ˆ ÿmf0 and then equation (I.11) will be written down as ~H ~ ÿ lH ~ ÿ mf. Then a straightforward computation from (I.7) – (I.10) yields the AˆD following system of equations

…I 12† :

A …Z † ˆ

4e 2 2 S …Z† ‡ 4aS…Z† ‡ …2k…m ‡ 1† ÿ l†Z ÿ 4Z…a†N ‡ hDH; Zix; m m

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…I 13† :

Pseudo-spherical and pseudo-hyperbolic submanifolds

A…N † ˆ ÿ4eDa ÿ 2e

2 m

525



tr…S2 † ‡ ma2 N

‡ e…ra ‡ …etr…S2 † ‡ …3m ‡ 4†k ÿ l†a†x

;

…I 14† :

A…x† ˆ k…DH†T

‡ k…Da ‡ a…etr…S2 † ‡ …3m ‡ 4†k ÿ l††N ÿ …2ema2 ‡ 4k…m ‡ 1† ÿ 2l ‡ km†x :

If we suppose that the mean curvature a is constant, then the above system reduces to

…I 15†

A …Z † ˆ

…I 16†

A…N † ˆ ÿ2e

…I 17†

A…x† ˆ k…etr…S2 † ‡ …3m ‡ 4†k ÿ l†aN ÿ …2ema2 ‡ 4k…m ‡ 1† ÿ 2l ‡ km†x :

:

:

:

4e 2 2 S …Z† ‡ 4aS…Z† ‡ …2k…m ‡ 1† ÿ l†Z; m m





2 tr…S2 † ‡ ma2 N ‡ …tr…S2 † ‡ …3m ‡ 4†ek ÿ el†ax; m

Now we are going to give some examples where these equations can be checked out. E x a m p l e 3 . Let Mnm be a minimal hypersurface of M m‡1 …k† such that the shape operator verifies S2 ˆ aI, a 2 R. Then one immediately checks that Mnm satisfies equations (I.15) – (I.17) with l ˆ 2…m ‡ 1†…k ‡ ea†; m ˆ 4ek…m ‡ 2†a and f0

ˆ m k‡ 2 I

:

E x a m p l e 4 (Generalized totally umbilical hypersurface). Let M m‡1 …k† be either the anti ‡1 …1†. Let or the De Sitter space Sm De Sitter space H1m‡1 …ÿ1† 1 m‡2 m‡1 …k†  Rs be a null curve with a local pseudo-orthonormal frame g:IR!M fA; B; Z1 ; . . . ; Zmÿ2 ; Cg tangent to M m‡1 …k† along g such that

hA Ai ˆ hB Bi ˆ 0 hA Bi ˆ ÿ1 ;

;

;

;

;

and g_ …s† ˆ A…s†; C_ …s† ˆ ÿk…s†B…s†;

for a certain function k…s† ˆ j 0. Then the map x : I 2 R 2 Rmÿ2 by x…s; u; z† ˆ f …z†g…s† ‡ uB…s† ‡

X

mÿ2 jˆ 1

zj Zj …s†;

f …z† ˆ

! M m‡1 …k†  Rms ‡2 given

q

1 ÿ kjzj2 ;

parametrizes a minimal Lorentzian hypersurface which is called a generalized totally umbilical hypersurface (it is called a B-scroll over g when m ˆ 2). It is easy to show that N …s; u† ˆ C…s† defines a unit normal vector field and the shape operator S associated to N

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ARCH. MATH.

verifies S2 ˆ 0. The example before says that this hypersurface satisfies that ~ ~ ˆ 2k…m ‡ 1†H. DH

! H31 …ÿ1† be the map defined by x…u v† ˆ …cos u cosh v sin u cosh v sin u sinh v ÿ cos u sinh v† It is easy to see that x parametrizes a minimal Lorentzian surface in H31 …ÿ1† that is called E x a m p l e 5 (Complex circle). Let x : R2 ;

;

;

;

:

complex circle (see [15]). A unit normal vector field is given by

N …u; v† ˆ …sin u sinh v; ÿ cos u sinh v; ÿ cos u cosh v; ÿ sin u cosh v†;

and the shape operator S associated to N has a matrix, relative to the usual basis of form  0 1 Sˆ ÿ1 0 :





x @x ; , @u @v @

ˆ ÿI and, again from Example 3, the complex circle satisfies   ~ ˆ ÿ12H ~ ‡ 16 f ‡ 1 I DH 4

So we have S2

:

E x a m p l e 6 (Non flat totally umbilical hypersurfaces). A non flat totally umbilical hypersurface is given by cutting M m‡1 …k† by a hyperplane in Rsm‡2 . Without loss of generality we canpchoice 2 the hyperplane P with them latter coordinate being k ÿ er , e ˆ 61, r > 0. Then Mn can be described by constant and given by p the set f…y; k ÿ er2 † : hy; yi ˆ er2 g, and therefore the quadric representation has matrix of form p2 ! yyt k ÿ er y G: f ˆ p t 2 k ÿ er y k ÿ er2 Since 2e…m ‡ 1† t yy ‡ C; D…yyt † ˆ r2 em Dy ˆ 2 y; r C being a constant matrix, we deduce that f is of 2-type and so equation ~H ~ ˆ lH ~ ‡ m…f ÿ f0 † holds, for appropriate l and m. D E x a m p l e 7 (Pseudo-Riemannian standard products). Let Mp …e1 r1 † 2 N mÿp …e2 r2 † be a pseudo-Riemannian product with e1 r12 ‡ e2 r22 ˆ k, where Mp …e1 r1 † and N mÿp …e2 r2 † are pseudo-spheres or pseudo-hyperbolic spaces according to ei ˆ 1 or ei ˆ ÿ1, respectively. Let x and y be the standard immersions of Mp …e1 r1 † and N mÿp …e2 r2 † into the corresponding pseudo-Euclidean spaces Rap‡1 and Rbmÿp‡1 , respectively. Then x 2 y is an isometric immersion of Mp …e1 r1 † 2 N mÿp …e2 r2 † into M m‡1 …k† whose quadric representation f ˆ …x 2 y†…x 2 y†t G has matrix of form fˆ

 xxt G

1

yxt G1

xyt G2 yyt G2



;

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Pseudo-spherical and pseudo-hyperbolic submanifolds

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G1 and G2 standing for the metrics on Rap‡1 and Rbmÿp‡1 , respectively. A straightforward computation yields 2e1 …p ‡ 1† t xx G1 ÿ 2Ip‡1 ; r2 e p1 e …m ÿ p† 1 D…xyt G2 † ˆ ‡ 2 r2 xyt G2 ; r12 2   e1 p e2 …m ÿ p† t D…yx G1 † ˆ ‡ yxt G1 ; r12 r22 2e2 …m ÿ p ‡ 1† t D…yyt G2 † ˆ yy G2 ÿ 2Imÿp‡1 : r22 D…xxt G1 † ˆ

~H ~ ˆ lH ~ ‡ m…f ÿ f0 † if and only if we only have two distinct eigenvalues. Hence Therefore D k ˆ e1 ˆ e2 and the radii are given by

…a†

r12

ˆ mp ‡‡ 12

and r22

ˆ mmÿ ‡p ‡2 1

…b†

r12

ˆ mp ‡‡ 22

and r22

ÿp ˆm m‡2

…c †

r12

ˆ m p‡ 2

and r22

ˆ m mÿ ‡p ‡2 2

;

;

:

References A. and P. LUCAS, 2-type surfaces in S31 and H31 . Tokyo J. Math. 17, 447 – [1] L. J. 454 (1994). [2] L. J. ALI´AS, A. FERRA´NDEZ and P. LUCAS, Hypersurfaces in the non-flat Lorentzian space forms with a characteristic eigenvector field. J. Geometry 52, 10 – 24 (1995). [3] M. BARROS and O. J. GARAY, A new characterization of the Clifford torus in S3 via the quadric representation. Japan. J. Math. 20, 213 – 224 (1994). [4] M. BARROS and O. J. GARAY, Spherical minimal surfaces with minimal quadric representation in some hyperquadric. Tokyo J. Math. 17, 479 – 493 (1994). [5] M. BARROS and F. URBANO, Spectral geometry of minimal surfaces in the sphere. Toˆhoku Math. J. 39, 575 – 588 (1987). [6] B. Y. CHEN, Finite-type pseudo-Riemannian submanifolds. Tamkang J. Math. 17, 137 – 151 (1986). [7] B. Y. CHEN, Null 2-type surfaces in Euclidean space. In: Algebra, Analysis and Geometry, 1 – 18. Taipei 1989. [8] B. Y. CHEN, Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169 – 188 (1991). [9] I. DIMITRIC, Spherical hypersurfaces with low type quadric representation. Tokyo J. Math. 13, 469 – 492 (1990). [10] A. FERRA´NDEZ and P. LUCAS, Classifying hypersurfaces in the Lorentz-Minkowski space with a characteristic eigenvector. Tokyo J. Math. 15, 451 – 459 (1992). [11] A. FERRA´NDEZ and P. LUCAS, On surfaces in the 3-dimensional Lorentz-Minkowski space. Pacific J. Math. 152, 93 – 100 (1992). [12] A. FERRA´NDEZ, P. LUCAS and M. A. MERON˜O, A quadric representation for pseudo-Riemannian product immersions. Tsukuba J. Math. 20, 435 – 456 (1966). [13] A. FERRA´NDEZ, P. LUCAS and M. A. MERON˜O, Classification of certain semi-Riemannian constant mean curvature surfaces. In: Geometry and Topology of Submanifolds, VII, F. Dillen et al., eds., 132 – 135. Singapore 1994. ALI´AS,

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[14] A. FERRA´NDEZ, P. LUCAS and M. A. MERON˜O, Biharmonic products in the normal bundle. Comment. Math. Univ. St. Paul. 45, 147 – 158 (1996). [15] M. A. MAGID, Isometric immersions of Lorentz space with parallel second fundamental forms. Tsukuba J. Math. 8, 31 – 54 (1984). [16] A. ROS, Eigenvalues inequalities for minimal submanifolds and P-manifolds. Math. Z. 187, 393 – 404 (1984). Eingegangen am 27. 11. 1995 Anschrift der Autoren: A. Ferra´ndez, P. Lucas and M. A. Meron˜o Departamento de Matema´ticas Universidad de Murcia 30100 Espinardo, Murcia Spain