Submanifolds of constant sectional curvature in pseudoRiemannian manifolds

( 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Submanifolds of Constant Sectional Curvature in

Pseudo-Riemannian Manifolds

JoAO

LUCAS BARBOSA, WALTERSON FERREIRA AND KETI TENENBLAT

Abstract: The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds Mn(K) of constant sectional curvature of pseudo-Riemannian manifolds Mn (K) of index s, with K K, flat normal bundle and such that the normal principal curvatures are different from K- K. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of R" which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of M, by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n - 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n - 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of M which have constant curvatures.

Key words: Constant sectional curvature, generalized equation, intrinsic generalized equa-tion

MSC 1991: 53C42, 53C40, 53B25, 53B20, 35Q20

1. Introduction

In [CT], it was shown that a class of equations called the generalized equation, is in-tegrable by providing its Backlund transformation and superposition formula. This class of differential equations for n x n matrix valued functions on n independent variables, includes the generalized wave and sine-Gordon equations and the general-ized Laplace and Elliptic sinh-Gordon equations as particular cases (see [G] for other generalizations).

the geometric interpretation of the whole class of equations called the generalized equation.

In this paper, we provide this interpretation in terms of Riemannian submanifolds of constant sectional curvature and we show that a submanifold which corresponds to a solution of the generalized equation is determined by its metric. Moreover, by considering a special class of solutions we obtain, among other results, manifolds of constant curvature K which are foliated by flat hypersurfaces with constant mean curvature and that are geodesically parallel (see [GG], [KW] for results on Rieman-nian foliations on compact manifolds of constant curvature).

We consider an n-dimensional Riemannian manifold Mn (K) of constant sectional curvature K isometrically immersed in a (2n - 1)-dimensional simply connected

-- 2n 1 -1

pseudo-Riemannian manifold M (K) of constant sectional curvature K, with index s, 0 < s < n - 1 and K K. We assume that the normal bundle of the immersion is flat and there is a point of M where the principal normal curvatures are different from K - K. Whenever M is a Riemannian manifold, i.e. s = 0, this is a necessary condition if K < K (see [Ca]). Moreover, the normal bundle is also necessarily flat, when K > K and the immersion has no weak umbilic points [M].

For such an immersion, we show that the structure equations, i.e. the Gauss and Codazzi equations, provide a system of second order differentiable equations which must be satisfied by an (O(n - q, q) matrix valued function, where < q n - 1 (Theo-rem 2.1). This class of equations is called the generalized equation due to previous denominations used in the particular cases considered in earlier works (s = 0, and q = 0 or q = n - 1). If we restrict ourselves to the differential equations which are satisfied by the metric of M, we have a system of equations, for a vector valued function, which is called the intrinsic generalized equation.

Solutions of the intrinsic generalized equation generically define Riemannian met-rics with constant sectional curvature, on open subsets of Rn, (see [ChT] for other equations) and are in correspondence with the solutions of the generalized equation.

-2n-1

In fact, any submanifold M' (K) of M -(K), as described above, provides a solu-tion of the generalized equasolu-tion and, hence, a solusolu-tion of the corresponding intrinsic generalized equation which is satisfied by its metric. Conversely, given a solution of the intrinsic generalized equation, which does not vanish, there exists a unique such submanifold, associated to the given solution, which is determined up to a rigid motion of M. This is proved in Theorem 2.3 and Corollary 2.4.

We observe that, besides its geometrical interest, the intrinsic generalized equation is also related to Hamiltonian systems of hydrodynamic type [DN].

Explicit solutions for the intrinsic generalized equation and for the generalized equation can be obtained by using the B5icklund transformation (see [CT], [T2]). This is a matrix Ricatti equation which can be linearized (see equation (28) and Theorem 2.2). We recall that the generalization, to higher dimensions, of the classical Bcklund transformation for the sine-Gordon equation was first obtained in [TT] by considering a beautiful geometric construction.

for n > 3. In [F], Ferreira provided an analogous result for the intrinsic generalized equations namely, it was shown that the symmetry group consists of translations, when K

#:

0, and translations and dilations when K 0.

Although it is possible to obtain many solutions of the intrinsic generalized equa-tion by the methods menequa-tioned above, it is a non-trivial problem (sometimes almost impossible) to obtain explicitly the associated immersed submanifolds. The difficulty consists on the integration of a system of differential equations whose compatibility conditions (the generalized equations) are satisfied. Examples of such immersions have been obtained explicitly in the special case in which the O(n - q,q) matrix associated to the n-dimensional manifold depends only on one independent variable (see examples and references mentioned in Section 2). In face of such difficulties, it is desirable to provide geometric properties of submanifolds associated to classes of so-lutions of the intrinsic generalized equation. This is the purpose of Section 3, where we consider those solutions which are invariant under an (n - 1)-dimensional group of translations. One naturally expects that the corresponding submanifolds possess a certain amount of symmetry. In fact we show that the submanifolds of constant curvature associated to those solutions are foliated by (n - 1)-dimensional flat sub-manifolds which have constant mean curvature (Theorem 3.6) and are geodesically parallel (see [F] for the case s = 0 and q = n - 1). Moreover, we prove that each leaf of the foliation is itself foliated by curves of M which have constant curvatures (Theorem 3.4). In Theorem 3.1 we provide an (n - 1)-parameter group of isometries of the submanifold M. Other geometric properties are described in Theorems 3.1, 3.5, 3.6 and Remark 3.7.

2. The Generalized Equation

We will denote by M'(K) a Riemannian n-dimensional manifold of constant

sec--2n-1

tional curvature and by Mn- (K) a simply connected pseudo-Riemannian manifold of constant curvature K and index s. We will use the following range of indices

1 < A, B, C < 2n - 1, 1 < i,j,k,< n, n +1 < a, /3 < 2n - 1.

-2n-1

-Let MS(K) be isometrically immersed in 1M, (K), where K ~ K and O < s < n - 1. Assume the normal bundle of the immersion is flat. We consider an adapted frame eA, such that e, 1 i < n, are tangent to M and (eA,eB) = aAJAB where aA = 1 except for a total of s indices, from n + 1 to 2n - 1, where CA =-1.

Let WA be the dual forms and WAB the connection forms defined by

2n-1

deA = OE (BWABeB. B=1

The structure equations in M are given by

2n-1

dWA = AW A BA, WAB + WBA = O, B=1

2n-1

dWAB = aUCWAC A WCB + QAB,

where

2n-1

QAB -=-2 rCUDRABCDWC A D.

C,D=1

If we restrict those forms to M we obtain w,, = O, i wi A wi, = 0 and

dwi =wjAwj (1)

It follows from Cartan's Lemma that

wia = bwj where b = bj. (2)

(2)

The Gauss equation is given by

dwij = wik A Wkj + Qij (3)

k

where

Qij

= C awi, A w,j + ij. (4)

By hypothesis QAB = -KaA BWA A WB, hence

Qij = -Kwi A wj. (5)

The submanifold M has constant curvature K if and only if

Qij = -Kwi A wj. (6)

Hence, it follows from equations (4)-(6), that M has constant curvature K, if and only if

ank-lWin+k-1 AWn+k-1lj = (K - K)wi A j. (7)

k=2

Since the normal bundle of the immersion is flat, we may consider the orthonormal basis ei such that bqi = 0, when i j (see for instance [DC], p. 137). Let

B(ej, ej) = wj,(ej)e,.

(8)

It follows from equation (7) that

(B(ej, ej), B(ek, ek)) = Zn+Y -i lbk- ' l =j: K -K Vk j. (9) i=2

The Codazzi equations are given by

dwic = wij Awja. (10)

Theorem 2.1. Let Mn(K) be a Riemannian manifold isometrically immersed in M2n- (K), where 0 < s < n - 1. Assume the normal bundle is flat and there exists a point p of M where the principal normal curvatures are different from K - K. Then there exist local coordinates xl,.. ., x, on a neighborhood of p such that, the first and second fundamental forms are given by

n n

I =a 2idx2 I=I K- i rJiiaijaljdX2.en+i (11)

i=l i=2,j=1

where a = 1 if K < K and a -1 if K > K; en+l,...,e2n-l is an orthonormal frame normal to M, aij are differentiable functions of Xl,..., Xn, and Jii is the

diagonal term of the matrix

n-q times q times

J-diag( ... 1, ,1,-1,. , 0O<q<n-n- 1 (12)

with q = s if K < K and q = n - (s + 1) if K > K. Moreover, the matrix function a = (aij) satisfies the following sytem of differential equations:

aJat = J

Oak akjhji, i :A j, hii =

Ohij + hj + hkihkj = -Kalialj, i j (13)

- hikhkj, i,j, k distinct OXk

where 1 < i, j, k < n.

Conversely, if a = (aij) is a solution of (12) and (13) defined on a simply con-nected domain Q C Rn, such that all, 1 < i < n, does not vanish, then there exists an immersion X : Q _ M n- 1 (K) which is unique up to a rigid motion of M, whose first and second fundamental forms are given by (11) and the index s is given by q if K < K and by n - q - 1 if K > K.

The system of equations (13) is called the generalized equation.

Before proving this theorem we observe that (13) is a system of second order differential equations for the O(n - q, q) valued matrix function a, which is reduced to a first order system by the matrix h defined by the second equation. Moreover,

when n = 2 and q = 0, by considering a = ( cosu/2 sinu/2) where u is a differentiable function of X1, 22, the system (13) reduces to

Uxlz1 - uX2X2 = -K sin u

which is the sine-Gordon equation when K 0 and the homogeneous wave equation when K = 0. Whenever n = 2 and q = 1, by considering a = (cosh u/2 sinh u/2

the system (13) reduces to

which is the elliptic sinh-Gordon equation when K 0 and the Laplace equation when K = 0.

The simplest example in the class of submanifolds considered in the above theorem is the n-dimensional Clifford Torus contained in the unit sphere S2 n - 1. In this case we have q = 0, K = O K= 1, and the matrix function a is a constant orthogonal matrix. When M and M are Riemannian manifolds with K

$

K, other examples are given by the so called toroidal submanifolds. These are generated by curves in such a way that each point of the curve describes an (n - 1)-dimensional torus [RT].

Example. An example of a hyperbolic 3-dimensional toroidal submanifold in R5 is generated by the curve

( |1,)D' b

where b h 0, c 0 0, D2 = 1 + c2- b2> 0 and v are elliptic functions of xz which are solutions of:

(Vl)2 = (b_ V2)(D _ V2),

2 ac2 b2 2

2 + C2 ( C2

1+ C2

i.e. the toroidal manifold is described by

(V2dx2 V2 CV3 cv3 nX

X(Xi, Z2, 3)= (J vldzl, Dcos x2, sin 2, - cos 3, sin

Example. The family of flat submanifolds of dimension n in the hyperbolic space H2n- 1 contained in the Lorentzian space L2n given by:

X = (vi + L, -aL, Xzlvl, a2vl - v2,...,bj cosxj, bj sin xj,...),j > 3

where

L(x, X2) = 2( 12+)v± 1 - EZV2,

and, vi are the following functions of x2:

V = eAcosh(2 - ) v2 = Asinh( 2 -5)

vj = bj, j> 3,

bj, A, E R, a2= 1, j=3 b = A2 -1, bj 0, 2C I and I is an interval which does

not contain 6.

Proof of Theorem 2.1. Since M is isometrically immersed in M and M is a Riemannian manifold, we may consider an adapted frame eA, 1 < A < 2n - 1, such that ei, 1 < i < n, are tangent to M and (eA,eB) = aAAB where ai = 1 for

1 < i < n and

if2jn-s

{ -_{_} 1 if 2< _{if n- s< <} <n- i K < K,

_if

_K

< nK,

and (14)

if2_<j_<s+l

nI

if{• s±1 < if K > K.

Moreover, since the normal bundle is flat we may choose the frame such that the dual forms wA and the connection forms wij satisfy equations (1)-(10). We observe that equation (9) can be written as

n

(B(ej,ej), B(ek, ek)) = n+i-lb nj+i-lbn+i-1 -crl- KI Vk j. (15) i=2

Now we consider J as in (12). By hypothesis, there is an open subset V of M such that, at each point of V, the principal normal curvatures are different from K - K. Therefore, since K L K, we may define alj on V by:

I o'lK Kj- - Ein2 orn+i- (bt?-1) 2

a1 lK- K +

K

i-2 ) In+-lb 1 < j < n, (16)

ijAjaI K - KI

where Aj = ±1 is chosen so that the right hand side is positive. We will show that d(wj/alj) = 0. In fact,

d( i )=d( 1 )A j +-dwj. (17)

ali alj alj

From (16) we have

n( o . n+i- 1 t'n+i-1

33 33 (18)

d(al) _ij ljaij E _Ti=2 vjj _{uiK - KI}l , (18)

On the other hand, from Codazzi equations (10) we get n

db+i- A wj + bnj+i-l dwj = _{33 kk} bnk+il _{Wjk A Wk} (19) k=l

Using (18) and (19) into equation (17) we obtain

n IW+i-lb+ -lbn+

d( ) -- jaj dW+j ±

n+i

J

w Akk A Wk (20)

From equations (15) and (1) we conclude that d(wj/alj) = 0 for all j, 1 < j < n. Therefore, on an open contractible region U of V, there exist smooth real valued functions x1, ...,

sx

such that

dx:j =-. (21)

It follows, from equation (1), that

wij = hijdxj - hidxi where hij -= 1 al (22)

ali 2xi

Now we define

a = bJ ila j/ -KI Vi > 2. (23)

Using (2),(21), (23) we have

in+k- = aki

K-

Kdxi

Vk,

2 < k < n (24)

and from (15) we obtain

n

oralialj + E an+k-lakiakj = for i

$

j. (25) k=2

Therefore, atJa is a diagonal matrix.

Substituting equations (22) and (21) into the Gauss equation (3), and collecting terms, we obtain the third and fourth equations of (13). Substitution of equations (24) and (22) into the Codazzi equation (10) produces the second equation of (13) for each k, 2 < k < n. In order to complete the system of equations (13) we only need to show that the coordinates can be changed so that the diagonal terms of atJa are those of J.

Taking derivative of (23) with respect to r,, we obtain using the second equation of (13) and (23)

b3 ai-1 (b n+i 1 -b aj i > 2 and j r.

aXr rr 33 i2,

Multiplying this expression by n,,+i-lbj+i-l and summing over i > 2 we get

((B(ej, ej), B(ej, ej))) alj = (B(ej, ej), B(er, er) - B(ej, ej)) 9a j r

It follows from Gauss equation (9) that

0a, ((B(ej,ej),B(ej, ej)) - (K -K) 2 ai

2j 9 r.

K - K - (B(ej, ej), B(ej, ej)) alj zxr Integration produces:

a2j IK- - (B(ej, ej), B(ej, ej))l = f(xj), 1 < j < n.

Hence, we can change each coordinate zj so that

a1j _{IK - El IK - K (B(ej, ej), B(ej, ej))j = 1 Vi.} K - - (B(ej, ej), B(ej, ej)) = V. (26)₍₂₆₎

IK

-KI

Let NpM be the normal space of the immersion at p. Consider on NpM R the product

Since K 0 K, ((,)) is a pseudo-Riemannian product which has index s (resp. s + 1) if K < K (resp. K > K).

Define

: TpMxTpM x NpMD R

(V, W) (B(V, W),(V, W))

We observe that f is a euclidean bilinear form [M] with respect to ((,)). Moreover, (26) implies that

alj 0(ej ej) alj 0 (ej, ej)

On the other hand, from equation (9) we have

(((ei, e),/~(ej, ej))) = O, i j.

Therefore, (ej, ej) form an orthonormal basis for NpM®9R, with the

prod-uct ((,)). Moreover, we can reorder ej so that for all j, 1 < j < n,

2~j( - Jjj _if K <

aIK j ((B(ej, ej), B(ej, ej)) - (K- Jjj if K>K.

Hence, using (2), (23) and (8) we obtain

k>2 {-Jjj if K > K.

As a consequence of (14), using the fact that at_{Ja is diagonal, we conclude that}

atJa = J.

The converse of the theorem follows from the Fundamental Theorem for

Subman-ifolds of pseudo Riemannian manSubman-ifolds. [

Solutions for the generalized equation can be obtained by a Backlund transforma-tion which shows that, for a given solutransforma-tion of (13), one can obtain a one-parameter family of solutions satisfying an initial condition. In order to give the Bcklund transformation, we rewrite the generalized equation in terms of exterior differential systems. We define the 1-form matrices

dzl 0

E = '.. , C = hE - Eht. (27)

E ( dx)n

Then the system of equations (2) reduces to

aJat = J daAE = aEAC

dC = CAC--EJAQE,

where

I is the identity matrix of order n.

Theorem A. Let Q C Rn be a simply connected domain and a = (aij) a solution of the generalized equation defined on Q. Then, for each constant z, z 0 if K 0, the initial value problem

dX + XJ-1/2CJ1/ 2 = JAzJ - XAtX

X(xo) = Xo, x0 E ,(BT(z))

where

Az = JMzaEJ-1 /2 _{Mz =}

zI - I) = diag(1,-1,...,-1)

has a unique solution X. Moreover, if Xo E O(n - q, q) then X E O(n - q, q) and it is a solution of the generalized equation.

The solution X given by the above theorem will be said to be associated to a by BT(z). The Backlund transformation given in Theorem A, was first obtained when q = 0, as a consequence of a geometrical result [TT].

The following theorem provides a superposition formula which shows that given a solution a of the generalized equation and a, a2 two solutions associated to a by

BT(zi) for constants z, i = 1, 2, z -7 z2, then a fourth solution can be obtained algebraically.

Theorem B. Let a be a solution of the generalized equation and let a, i = 1, 2, be solutions associated to a by BT(zi), zl z2. Then there exists a solution a*,

associated to a and a2 by BT(z2) and BT(zl) respectively, given algebraically by

the expression

a*Jat = J(alJaMz, - JMZ2)- (JMZ, - alJatM2).

Although the solutions given by Theorem A may be complex valued, whenever q 0, by using conveniently the superposition formula, one can obtain, from a given real solution, a family of real solutions of the same equation. This is the content of the next result.

Corollary. Let a be a real solution of the generalized equation and let a be a solution associated to a by BT(zl). Then the real valued matrix function a*, given by

a*Jat = (alaitMz,, - Jz,)-l(alatMz2 - JMz),

is another solution of the generalized equation, where ial and M, are the complex conjugate of al and M1,, respectively.

the above theorems were given in [T2], where one and two soliton solutions for the generalized equation with q = 0 were explicitly given. Such solutions were obtained by applying Theorem A to the trivial solution a = I and then applying Theorem B.

Example. In particular for n = 2, K = -1, K = 0 we obtain the following solu-tion for the generalized equasolu-tion, associated to the identity solusolu-tion by a Bicklund transformation

a= ( sin cos sin ) where u = 2 arctg e.

The surface that corresponds to this solution is the pseudosphere.

Similarly, we consider equation (13) with n = 2, q = 1 and K = 1. We start with the trivial solution a = I. Applying Theorem A we obtain a one-parameter family of complex valued solutions of (13), associated to a by BT(z), given by

1 (l+f2 2f )

al = 1_ f-2 - 2f 1+ f2 ,

where

f(Xl, X2) = exp(bxl + icz2), b = (Z - z-1), c = -(z + z-1), z E R \ {0}.

The corollary provides real solutions given by

a = 1 ( l+g 2 2g w bcosCX2

1 g2 2g g(, 2 , where+ = c cosh bzl

The matrix Ricatti equation BT(z) can be linearized by considering X = PQ-1, where P and Q are (n x n)-matrix valued functions. This leads to the linear system

JAZJ P

d Q

=

L

At J-CJ2

Q

The compatibility conditions for this system with parameter z are the same as those for BT(z). Thus the generalized equation implies the existence of a fundamental

matrix solution

d [ J- CJ Ibb (28)

with g0 an invertible (2n x 2n)-matrix valued function. This equation implies that det 0 is constant, so we may assume that det 0 = 1.

The relationship between solutions of BT(z) and solutions of (28) is the following.

Theorem 2.2. Let a = (aij) be a solution of the generalized equation. For each z, let 0 be a solution of (28) such that det _ 1. Consider constant (n x n) matrices ml, m₂ and set

[ QI= [ m2 (29)

If Q is invertible and X = PQ-1 has values in (9(n - q, q), then X is associated to a by BT(z). Conversely, for any X associated to a by BT(z), there are matrices

ml, m2 such that X =

The proof of this result follows the same arguments used in [BT1], for the particular case K < K and q = 0. We observe that in view of Theorem 2.2, applying the Bicklund transformation, corresponds to solving the linear system (28).

Now we introduce the intrinsic generalized equation, which is a system of equations for a pair v, h}, where v = (vl,..., vn) and h is an off-diagonal (n x n)-matrix

function defined on Q C Rs, which satisfy

vJvt =1 dvi

a9i = vjhji - JiiSij E Jssvshis

xj s>1

ahi + 9hj E hh = -Kvivj, i j, (30)

19xi 1xj ,Sij

hij = hishsj, i,j,sdistinct, aX

hJii + J jj- +

Z

J,,shihj = , i j, 3 sfi,j

where 1 < i, j, s < n.

Remark. Whenever v does not vanish on an open subset U C Rs, the second equation for i = j and the last equation of the system above are consequences of the other equations. Moreover, v defines a Riemannian metric on U with constant sectional curvature K, since the third and fourth equation of (30) is the Gauss equation. As in the case of the generalized equation, when n = 2 the above equation also reduces to the wave equation (q = 0, K = 0), the sine-Gordon equation (q = 0, K

$

0), the Laplace equation (q = 1, K = 0) and the elliptic sinh-Gordon equation (q = 1, K & 0).

The intrinsic generalized equation, can be written in terms of exterior differential

systems as

vJvt = 1 dv = -vBJ

dC = CAC- - EJAQE (31)

2 dB = BJAB.

where C and E are given by (27) and

B = JhtE- EhJ. (32)

There is also a Bcklund transformation for the intrinsic generalized equation which can be found in [CT]. Our next result relates solutions of the generalized equation and the intrinsic generalized equation, generalizing previous results (see [BT2], [C2]).

(ii) Conversely, if v is a solution of the intrinsic generalized equation whose coordi-nate functions do not vanish on a simply connected domain Q C Rn, then there exists a solution a on Q for the generalized equation whose first row is v.

More-over, if a and are any two such matrices, then a = PJa, where P = ( W)

is a constant matrix, with W E O(n - q - 1, q).

Proof. If a is a solution of the generalized equation we consider v with coordinate functions vj = aj. It follows from Remark 2.1 that v satisfies the intrinsic gen-eralized equation. In case K = 0 we can also consider vj = arj for any fixed r,

r > 2.

Conversely, suppose v is a solution of the intrinsic generalized equation whose coordinates do not vanish. Let h be an off-diagonal matrix function determined by v using the second equation of (30). Then B defined by (32) satisfies dB = BJ A B. Therefore, the system of equations

dy = -yBJ

has a unique solution y: Q - Rn having a prescribed value at any given point of Q. Moreover, yJyt is constant since B + Bt = 0. Hence, there exists a fundamental matrix solution a for da = -aBJ, such that aJat = J and aj = vj. Therefore, a satisfies the generalized equation.

If is another such matrix then it follows from the skew-symmetry of B, that P = Jat is a constant matrix. Since pjpt = J, we conclude that a = PJa where

P satisfies the required conditions. [

The above result shows that any Riemannian manifold Mn(K) isometrically im-mersed in a pseudo-Riemannian manifold M2n- (K) with K K, associated to a solution a = (aij) of the generalized equation, is determined, up to rigid motions of M, by its metric which is a solution of the intrinsic generalized equation. In fact, it follows from Theorem 2.3 (ii) that the second fundamental form, which is given by the last n - 1 rows of the matrix a, is uniquely determined up to a change of normal frame. Therefore, we have

Corollary 2.4. A Riemannian manifold M'(K) isometrically immersed in a pseudo-Riemannian manifold M2n-1 (K) with K T- K, flat normal bundle and such that the principal normal curvatures are different from K - K, is uniquely deter-mined, up to rigid motions of M, by a diagonal metric gij = 6ijvi2 where v satisfies (31).

We have already mentioned that solutions for the intrinsic generalized equation can be obtained by using the Backlund transformation. One can also obtain explicit solutions for this equation by considering those solutions which are invariant under the symmetry group of the equation. This group is given in the following result.

Theorem C. The symmetry group of local Lie-point transformations of the intrinsic generalized equation for n > 3 is given by

= ebxi + ai

h = e-bhij

where ai, b E R and b = O when K 0.

The proof of this result for the case q = 0 was first obtained by Tenenblat-Winternitz [TW] and in the general case by Ferreira [F]. Several explicit solutions invariant under the symmetry group in terms of elliptic functions can be found in [C1], [F], [RT]. As an illustration we give the following example.

Example. We consider the solutions of the intrinsic generalized equation, for n = 3, q = 2 and K = -1,0 or 1 which are of the form v((), where = lcizi, ai E R\{O}, such that none of the coordinate functions vj are constant. Then the solutions are given by

(v,b)2 b(a- b)(v2- 1)(v2- - b)

V2 a-b 2 a-c

2 a-(

3 a "v a - b'

where a, b are distinct, nonzero real numbers, c E R and satisfy the relation

(a - b)(bao + aao2) + abao + K = O.

3. Geometric Properties of the Submanifolds Associated to the Invariant Solutions

In this section we obtain some geometric properties of the submanifolds associated to the solutions of the intrinsic generalized equations which are invariant under an (n - 1)-dimesional group of translations. Such solutions are of the form

V() =(1 Vn())n(),

where

n n

=-ci Xi, oi E R, and i 2 0.

i=l i=l

We will consider solutions defined in an open simply connected domain Q C Rn. Since the set of EE R such that vi(E) O Vi, 1 < i < n, is an open subset of R we will assume in what follows that Q is of the form

n

Q = {x E Rn : 1 <

cixi

< E2}

i=1

where E2 E [- o, o] and 1 < '2.E1,

Let V and V be the Riemannian connections of M = X (Q) and M. Then X satisfies the following equations

VxjXi- E1kj Xk + aij [/j- IE Jsaliasi en+,- (33)

k s>2

Vxje+-s- = -- -[jK IKIXj (34)

where X = and

Qi

are the Christoffel symbols of the connection V given by

rk. = , i, j, k distinct,

rj = ajvi< (35)

vi

Vi, Vi

Pi = -i 2 2

We observe that rFki depend only on .

Theorem 3.1. The Riemannian manifold (Q, g), where the metric g is defined by gij = 5ijvi2(1), has the following properties:

a) The sectional curvature of Q is constant and equal to K.

b) The hyperplanes of Q defined by

Zt=

aixi = o, I < < 2, with the in-duced metric are fiat, have constant mean curvature (depending on o) and are geodesically parallel. Moreover, if K < 0 the hyperplanes cannot be minimally immersed in Q.

c) For any set of vectors U1,...,U,-1 generating the hyperplane E= l oaiXi = 0

the (n - 1)-parameter group of translations T,: Q - Q defined by

T:(z) = +U 1 + + '-1Un-l,

where = (El, . ,-1) E Rn - l, are isometries of Q.

Proof. Since v is a solution of (30), (Q, g) has constant sectional curvature K and the metric gj restricted to the hyperplane Po: aiml cixi = o is constant. It is easy to see that there exist coordinates in Pg for which the coefficients of the metric are constant. Therefore, the curvature tensor is flat.

Let f(z) = cEn=l ixi, e Q. In order to prove that the hyperplanes P'0 = f-l (o)

are geodesically parallel, it is sufficient to show that grad f(x) is a non zero constant along each hyperplane. In fact, we have grad f(x) = E i 1aiv-2 ( )aO/dxi and hence

[grad f(x)l = v ) 2

is a nonzero constant along the hyperplane PCO. We conclude that the trajectories of the vector field -grl are geodesics of Q and Po are geodesically parallel.

Let x E P6 and A: TxPCo -+ TxPo0 be the adjoint operator associated to the

second fundamental form of Pg0, i.e.

From the definition of f, we have the Laplacian of f given by

Af= akrki

i k=1 vi( 0)

Since Af and I grad f are constant along P~0 it follows that

trace A =- Af (x) I grad f(x)I

is constant along P,0. Therefore, we conclude that PC0 has constant mean curvature. In order to complete the proof, we observe that

(n - 1)2H2 = IA12 - (n - 1)(n - 2)K,

where H = trace A/(n - 1) and A12 - E7n-₁ (A(Ei),Ej)2

- .i,j={A(Ei), Ej}2 and E, 1 < <

i

-1

is an orthonormal basis of P0. Hence, H f 0 whenever K < 0. O

Our next result shows that the immersion X: Q - M takes each line of the hyperplane i 1aixi = o into a curve of constant curvatures in M. In order to prove this result we need two technical results.

We observe that since the metric is a function F the coefficients of the second fundamental form are not necessarily a function of J only. However, we will show that some products of those coefficients depend only on . In order to state the result we denote by p = (Pl, P2,. . .,P) Zn, Pi > 0, 1 < i < n, an ordered multi-index and by IplI = ¥£ ₁Pi the order of p. Moreover, the differential operator aP will denote

IPI

1

. . .

n

Lemma 3.2. Let Q C Rn be an open set and Zi: - Rn-1, 1 < i < n, the function defined by Zi = (a2i, a3i,..., ani), where aji, j 2, are the coefficients of

the second fundamental form of the immersion. Then the product

n

O (fZi) aOq(gZs) = E JrrP(fari)0q(gars)

r=2

depends only on , V, q E Zn and f, g: Q - R C°°-functions of .

Proof. We will first prove the lemma for the particular case f = g 1, by using induction on IPI +

q].

If IPI + Iq = 0, then

n

*PZi OqZs = Zi . Zs E ajiJjjajs = -Jllalials + JiiSi j=2

which depends only on

C.

If

Ipl

+

ql

> 1, we assume that for all multi-indices j3 and q, with 0 < I +

Il

< Ipl + Iql - 1, opZi PqZs depends only on ~. Without loss of generality we may assume p > 1. Then

for some 1, 1 < I < n, where is of order p - 1. From the second equation of (30) we have that

_Zi - hliZl if

l

i

ax

aZi

=

-

J

Eri Jr,hirZ

Therefore,

F

,

(Pa)

(ahD(ap--Zi) .YaZ8 if 1 i,

PZi · qZs = (36)

-Ji

E

Jrr

)

(Vhr)(P Zr) agZ,

if

i =i

where, p (P1, ..., /n) is a multi-index and P <

P

means pi < fii, i, 1 < i < n. Moreover, the multi-binomial () denotes the product () = (P1 )(02) .. (p) and

]i<g stands for

=o

₀ E2=0 ... =

Using equation (36) and the induction hypothesis, it follows that aPZ,· dqZs depends only on . The general case follows from the case f = g _ 1 and the

Leibnitz rule. Cl

The second technical lemma provides an expression for the successive covariant derivatives of equation (33). If I = (i1, ..., i) is an ordered multi-index of integers

with I > 1, and Y is a vector field along M = X () we will use the following notation

VjY = Vx, " Vxj2Vxj, Y,

aj(f)=

a9rf

axil...Oxi

'

Lemma 3.3. If I = (il,..., i) is a multi-index with I > 1, then

VXj

=

E/ f

jXk

+ E Es~je+sj i (37)

k s>2

where

1-1

Es,j= Oir(Hrj,Jssali,-rasi,-r), Ir = (il-r+l, il-r+2, ... ,il), 1 < r < -1,

r=O

0Io is the identity operator, flkj and Hrj are real functions of .

Proof. The proof will follow by induction on 1. If I = 1, i.e. I = (i1), then equation

(33) reduces to (37) with

Enj = lij

J s

salilasil,

f 1 A = , rkl j and Hlj = il>j V - K I. Hence the lemma holds for I = 1. Assume that it holds for I > 1. Let I = (il, ., il il+1) then

Observe that using the induction hypothesis and equations (33) and (34) we have

E f/ijVX +l, Xk = fXjrkil + E5 f. Jssalij+l K asil+l en+s-l

k r,k s>2

and

E El,jVxt+l en+s-1 = G,jXil+l s>2

where

1-1

Gf j _ E Zi+ 1 9I, (HjjajirZi) K - /

KI

alit+l r=0

and Zr = (a2r, ... , anr) is given as in Lemma 3.2. It follows from this lemma, that GIj is a function of I only.

Using the induction hypothesis and subtituting the two last expressions into (38) we obtain that

VIXj = fr jX + E e+s-l,7

s>2

where

f' -- i,+J - kJ,j' kil+l - Sri,+lGIj

and

ESj = E AiJs (Hlr,jalii+l-asi+l-+-) ,

r=O

with H j = fIj'+l K - and H = Hr , Vr, 1 < r < l. Therefore, the lemmaIr,

holds for I. [

Theorem 3.4. Let q be any point of Q and U a vector of the hyperplane aE=i

oixi = 0. Then

a) y(t) = X(q + tU), t E R, is a curve of M2n - 1(K) of constant curvatures. b) y(t) is congruent to y(t) = X (+tU) if q and q are points of the same hyperplane

n=1 aiXi = 0.

Proof. We consider q = (ql,..., q) and U = (u,..., n). Let o be defined by

ZEm=1aiqj = o. Since

y

= uxj

(39)

we have that 1-y'l2 = Ej uj2jV(o) is constant. We will show that D%' is also constant Vr > 1, where D denotes the covariant derivative of M. This will imply that all the curvatures of y are constant.

dtr - ujuiViXj

j,I

where I = (il, . . ., i) and ul = il... Ui_r.

Let I = (il,..., i,) and I' = (i, ... , i) be two multi-indices of order r > 1. From equation (37) we obtain

(VXjV, Vl·,Xj ) = fkj,,j(k) 2

k r-1

+ Oi, (H1,jalj,_ Z -_i)·(HTja Z

ij

We conclude that 'I is constant by using Lemmas 3.2 and 3.3. Moreover since this constant depends only on o and the vector U, it follows that if q and are points of the same hyperplane Ei=1 aixi = Eo, then the curves y(t) and y(t) = X( + tU)

have the same curvatures, therefore they are congruent. [1

Particular cases of the following result were obtained in [Am], [C1], [DT], and [RT].

Theorem 3.5. Assume that j = 0 for j L = {Jl,...,jm}, for some m,

1 < m < n. Then there exist m orthogonal vectors Uj, j E L of the hyperplane E=l °ii x= 0 such that for each q E Q the curve yj(t) = X(q + tUj), t E R, where j G L is contained in a two dimensional totally geodesic submanifold of M. Moreover,

yj has nonzero constant curvature.

Proof. Let q = (ql, ..., q) be the point in Q. For each j E L we consider Uj to be the j-th vector of the canonical basis of Rn . Then -yj(t) = Xj(q + tUj). Hence,

YjI

= Ivj(o) where vj(l) is a solution of the intrinsic generalized equation and ° = Z 1aiqi.

Since

D.

= VxjXj, it follows from equation (33) that the first curvature of yj is a nonzero constant. Moreover, since a,j does not depend on x, for r L we obtain

dt2 = Vx V7xjXi =~ E I (a)2Js l -IXj, j L

kdL,I s>2

Here we have used the fact that the metric and the Christofffel symbols are constant along the hyperplane i=l ciqi = Eo.

Using equations (33)-(35), since j E L, we conclude that

dt2 (a

rjk

+ (Jjj -Jaj)K - KI j

It follows that the second curvature of yj is identically zero. Therefore, 7j is con-tained in a two-dimensional totally geodesic submanifold of M. [

Theorem 3.6. Let NnO'-1 = X(Po) where 7P~o is the hyperplane of Q given by

in=

1oxi =0o. Then

b) The principal normal curvatures of M, (B(ei, ei), B(ei, ei)) where ei = Xi, are constant along NEo. Therefore, the norm of B is constant along No.

Proof. a) follows from the properties of P&0 given by Theorem 3.1. In order to prove b), we observe that

(B(ei, e), B(ei, ei)) =

K

(Jallai - Jii)

Therefore, it is constant along N~. [

Remark 3.7. Given a solution of the intrinsic generalized equation of the form v(() where = =1 oLixi, the associated constant curvature submanifold X(x, ..., x,) can

be reparametrized as follows. Consider n-1 linearly independent vectors U1, ..., U_1 of the hyperplane E=1 eizi = 0 and let Uo = (a,., ,)/ 2. ?l Then each point

p = (l,..., zx) E Q is uniquely written as

n-l

p= Uo+ XiU, < < 62, Ai E R.

i=l

We define

n-1

Y(, A

1

,

...

, A,-1) = X(EUo +

AiUi).

i=l

Then for each o, { < o < 62, Y(o, Al, ..., ,_n-) is an (n - 1)-dimensional flat submanifold of constant mean curvature and the coordinate curves

Ai -- Y(o, A°_{, ...Ai, ..., A,_}

1) are curves of constant curvature in M.

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J.L.M. BARBOSA W. FERREIRA

Departamento de Matemtica IMF-Departamento de Matemtica Universidade Federal do Ceari Universidade Federal de Goias

Campus do Pici Campus II- Samambaia

60455-000 Fortaleza, CE 74000 Goiania, GO

Brazil Brazil

K. TENENBLAT

Departamento de Matemtica Universidade de Brasilia 70910-900 Brasilia, DF Brazil