REND. SEM. MAT. UNIVERS. POLITECN. TORINO

Vol. 44#, 2(1986)

JohanDeprez (*)

SEMI-PARALLEL HYPERSURFACES

Abstract. We define semi-parallel submanifolds as extrinsic analogue for semi-symmetric spaces and give a local classification of semi-parallel hypersurfaces of Euclidean space.

1. Introduction and statement of results.

Let (M,g) be a (connected) Riemannian manifold and denote by V its Levi Civita connection and by R the Riemannian curvature tensor. Semi-symmetric spaces, i.e. Riemannian manifolds for which RmR = 0, were first investigated by E. Cartan [2]. By RR = 0 we mean that R(X, Y) R = 0 for all tangent vectors X and Y to M where the linear endomorphism R(X, Y) acts as a derivation on R. Semi-symmetric spaces can be seen as a direct generalization of locally symmetric spaces, i.e. Riemannian manifolds which satisfy VJR = 0. More recently, Z.I. Szabo obtained a local classifica­tion of semi-symmetric spaces [9]. K. Nomizu studied semi-symmetric hyper­surfaces of Euclidean space [6] and P.J. Ryan classified semi-symmetric hypersurfaces in real space-forms 17] and semi-symmetric complex hypersur­faces in complex space-forms [8]. For references on hypersurfaces satisfying similar conditions on other curvature tensors, one can consult [3].

Let i : (M,g)—>Em be an isometric immersion with h as second fun­damental form and V as connection of can der Waerden-Bortolotti. The

Classificazioneper soggetto AMS (MOS, 1980): 53B25

(*) Aspirant N.F.W.O. Belgie.

304

notion extrinsic locally symmetric immersion (also called parallel immersion), i.e. an immersion with Ih = 0, was introduced by D. Ferus in [4] as an extrinsic analogon for locally symmetric spaces. In the same paper Ferus provides a classification of this type of submanifolds. For references on extrinsic locally symmetric immersions, see [4].

We define semi-parallel immersions to be immersions which satisfy the condition R • h = 0, where R denotes the curvature tensor of the van der Waerden-Bortolotti connection. In this paper we will prove the following classification theorem for semi-parallel hypersurfaces of Euclidean space.

THEOREM l. Let i: Mn—• En+1 be an isometric immersion of an n-di-mensional Riemannian manifold as a hypersurface of En+1. i is semi-parallel if and only if it is one of the following immersions:

(i) an open part of a standard imbedding of a hypersphere Sn of En+1, (ii) an open part of a standard imbedding of an elliptic hypercone Cn of

Ew + 1 ,

(iii) a cylindrical immersion,

(iv) a locally extrinsic product of a standard imbedding of an nx-sphere in Eni and an (n-nx)-plane Enni (nx G {2,. . . ,» — 1}),

(v) a locally extrinsic product of a standard imbedding of an elliptic nx-conein Eni and an (n-nx)-plane En ni (nx E {2, . . . ,» - 1}).

An immediate consequence is the following.

COROLLARY. Let i: Mn —> En+1 be an isometric immersion of a complete n-dimensional Riemannian manifold as a hypersurface of En+1, i is semi-parallel if and only if it is one of the following immersions:

(i) a standard imbedding of a hypersphere Sn of En+1, (ii) a standard imbedding of a hyperplane En of En+1,

(iii) a hyper cylinder, i.e. the extrinsic product of a complete plane curve with an (n- 1)-plane,

(iv) a spherical hyper cylinder, i.e. the extrinsic product of a standard im­bedding of an nx-sphere in Eni and an (n-nx)-plane (nx€z G {2,... , I I - 1 } ) .

It is easily seen that the immersions in (i) and (iv) in Theorem 1 are parallel while those in (ii) and (v) are not and those in (iv) are parallel only if they are an open part of a hypercylinder over a circle.

305

The proof for Theorem 1 also serves as a proof for the following local version of a theorem of Nomizu [6J.

THEOREM 2. Let i: Mn —• En+1 be an isometric immersion of an n-dimen-sional Riemannian manifold as a hypersurface of En+1. Mn is semi-sym­metric if and only if i is one of the following immersions:

(i) an open part of a standard imbedding of a hypersphere Sn of En+1, (ii) an open part of a standard imbedding of an elliptic hyper cone Cn of

En + 1, (iii) an immersion with type-number at most 2 in every point, (iv) a locally extrinsic product of a standard imbedding of an nx -sphere in

Eni andan (n -nx)-plane En~ni (nx G {3,.. . ,n- 1}), (v) a locally extrinsic product of a standard imbedding of an elliptic nx-

cone in Eni andan (n — n1)-planeEn~ni (wt G {3,.. . ,w — 1}).

REMARK. For immersions with type-number at least three in one of their points, semi-symmetry implies semi-parallellity.

We also investigated immersions of Riemannian manifolds as hypersur-faces in Euclidean space that satisfy C'A = 0, where C denotes the Weyl conformal curvature tensor of the Riemannian manifold and A the second fundamental tensor of the immersion. We obtain the following theorem.

THEOREM 3. Let i: Mn—• En+1 be an isometric immersion of an n-di-mensional Riemannian manifold as a hypersurface of En+1 (n>3). The following assertions are equivalent:

(i) C(X, Y)'A = 0 for all tangent vectors X and Y,

(ii) Mn is conformally flat.

2. Basic formulas.

For later use we state some results of Z.I. Szabo [9]. Let (Mn,g) be a C~ Riemannian manifold and p GMn. Let TpM = V<0)(p) + ^(p) + ... + + V^rHp) be the irreducible decomposition of the tangent space with respect to the group "Kp, associated to the Lie subalgebra k(p) := alg {R(X, Y) | X, Y G ETpM] of 0(TpM). Thus the subspaces V^^ip) are invariant under the action of JCp; they are pairwise orthogonal. Furthermore Kp acts on V^°Hp)

306

trivially, and its action is irreducible on V^*Hp), i>0. This decomposition is called the V-decomposition of the tangent space.

THEOREM (first stability theorem) [9]. For a Riemannian manifold (Mn,g) there can be chosen a dense open subset U of M such that on every arcwise connected component Ua of U for a certain order V*0)(p), ^^(p) , . . . , V^Hp), P^UOL °f the invariant subspaces the subspaces V^fHp)} pEUa, have constant dimension, and the distributions p)—• V*fHp) are of class C00 for any index i. Such a differentiable decomposition TM'..= V^ + + V^l) + ... + V^ of the tangent space is unique in the sense that if on an open set Q the decomposition TM = V<0) + ^ 1 } + ... + V<r) 'is of class C°°, then the C™-distributions V^ are unique on Q up to an order.

Define subspaces Z^(p) := vect { Vx ^ VX2... Vx« X% | Xj vectorfields with

values in F ^ for l < / < £ , £ G N 0 } for i € { l , . . . , r } . Let Z{i)(p)D fiZ(»(p) = {0} for i¥=j and ^ ) C Z ( i > ( p ) C ^ ( p ) + ^ ( p ) . Let Zi0)(p) : = (Z(1)(p) + ... + Z^(p))L. The decomposition TpM = Z(0)(p) + + Z^(p) + ... + Z^ip) is called the Z-de composition of the tangent space.

THEOREM (second stability theorem) [9], The subspaces Z(l)(p), p€U, r are pairwise orthogonal, and there can be chosen a dense open subset G of Mn (which is also a subset of U) on the arcwise connected components of which the subspaces Z(,)(p) for any index i have constant dimension and the distributions p f—• Z(,)(p) are of class C°° in G.

It is known that the distributions Z(,) are totally parallel on G and that they are thus involutive. Call the integral manifolds Af,. M0 is of zero curvature and M\,...,Mr are irreducible.

A Riemannian manifold (Mn ,g) is called a simple leaf if at any point the V-decomposition of the tangent space is of the form TpM = V^°^(p) + + V^^ip). A simple leaf is said to be infinitesimally irreducible if at least at one of its points the Z-decomposition contains only the subspace TpM = = Z(1)(p). So we can state the following theorem.

THEOREM (local decomposition using the local or infinitesimal holonomy group) [9]. For any C00 Riemannian manifold (Mn ,g) there exists a dense open subset G such that around every point p^G the manifold can be decomposed into a direct product of Riemannian manifolds in the form MQxMx x ... xMr, where M0 is of zero curvature and furthermore the

307

manifolds Mi, i > 0, are infinitesimally irreducible simple leafs.

Let i: (M,g)—• (M,g) be an isometric immersion of an ^-dimensional Riemannian manifold in an w-dimensional Riemannian manifold. Let £ be a local unit normal section on M. In the following X, Y,Z,U, V denote vectorfields which are tangent to M. Then the formulas of Gauss and Weing-arten are given by Vx Y = Vx Y + h (X, Y) and Vx £ = - A%X + Dx {, where V is the Levi Civita connection on (M,g), V the Levi Civita connection on (M,g) and D the normal connection of the immersion (we identify TM and i*(TM)). The second fundamental tensor A^ is related to the second fundamental form h by g(h(X, Y), £) =g(AtX, Y).

Denote by R the curvature tensor of V. (M,g) is called a. locally sym­metric space if VR = 0 and a semi-symmetric space if R (X, Y) • R = 0 for all X,YG 3C(M). . i is parallel (or locally extrinsic symmetric) if lb = 0, i.e. Dxh(Y,Z)-h(\/xY,Z)-h(Y, VxZ) = 0 for all X,Y,Z£3C(M).

DEFINITION. An isometric immersion i: (M,g)—>(M,g) is semi-parallel if fl(X,r)/b_: = Vx(Vy/?)- VY(Vxh)- \x>Y]h = 0, when # is the curvature tensor of V.

If i*1 denotes the curvature tensor of D, then (R(X, Y) 'b)(U, V) = = Ri(X,Y)h(U,V)-b(R(X,Y)U,V)-b(U,R(X,Y)V). In terms of the second fundamental tensors, an immersion is semi-parallel if and only if ARix v - [ ^ ( ^ ^ ) ^ | ] = 0 f°r all X , 7 G f ( M ) and all vectorfields

£ normal to M. Let ij : (MnJ,gj)—*(Mmi,gj) be isometric immersions of Riemannian

manifolds in Riemannian manifolds (1 < ;<&) , The extrinsic product of ix,..., ik is the isometric immersion

f i x ... x ik : M"1 X ... x Mnk —• M™1 x ... x Mmk

(xu...,xk) i—• ( f ' i ( * ! > , . . . , ! * ( * * ) )

with the product metric. An isometric immersion i: (M,g)—>(M,g) is a locally extrinsic product of ii,...Jk if and only if i is locally congruent to ij x ... x ifr.

Let i: (Mw,g)—• 2?w+1 be an isometric immersion of a Riemannian manifold as a hypersurface of En+1. Let J be a local unit normal section on M. We denote 4̂t by A and consider the scalar-valued second fundamental form a determined by Hx Y = Vx Y + o(X, Y) £.

308

Let XAY denote the endomorphism Z +~+g(Z,Y)X-g(Z,X)Y. Then the curvature tensor R of M is given by the equation of Gauss

R(X,Y) = AXhAY. (1)

The Codazzi-equation becomes

(VXA)Y=(VYA)X. (2)

Since A is symmetric there exists an orthonormal frame eit...,en consist­ing of eigenvectors, i.e. such that Aei = \ei1 ' (* G {1,2,...,»}), where X!,X2,...,XW are the principal curvatures of the immersion, i is called cylindrical if at most one of the principal curvatures is non-zero. The number of non-zero eigenvalues at p EMn is called the type-number of the immer­sion at p.

(AA",g) is called locally conformally flat if and only if for each point x in AP there exists a neighbourhood U of x, an open subset V of En

and a conformal map 0: V —• U. Denote by C the Weyl conformal curva­ture tensor of (Mn,g). We recall that every surface is conformally flat and that for every 3-dimensional Riemannian manifold the Weyl conformal curvature tensor C vanishes identically. It is well-known that for dim M > 4 M is conformally flat if and only if C = 0. i: (Mn,g)—>En+1 is said to be quasiumbilical when M has a principal curvature with multiplicity > » — 1. For dim M>4 E. Cartan proved that (Af,g) is conformally flat if and only if i is quasiumbilical [1].

3. Lemmas.

LEMMA l. Let i: (Mn,g)—>En*P be a semi-parallel immersion. Then Mn

is a semi-symmetric space.

Proof. If R, respectively R^- denotes the curvature tensor of Mn, respec­tively the normal curvature tensor of i, < . , . ) the metric of En+p and h the second fundamental form of i, then

g((R{U,V)'R)(X,Y)Z,W) = = g(R(U>V)R(X,Y)Z,W)-g(R(X,Y)R(U,V)Z,W)-g(R(R(U,V)X,Y)Z,W)

-g(R(X,R(U,V)Y)Z,W)

309

= (h(X,Z), h(Y, R(U, V) W)) - <b(X, R(U, V) W), h{Y,Z)) -<h(X,W),h(Y,R(U,V)Z))

+ <h(X,R(U,V)Z),h(Y,W))-<b(R(U,V)X,W),h(Y,Z)) + (h(R(U,V)X,Z),h(Y,W))-<h(X,W),b(R(U,V)Y,Z)) + <h(X,Z),h(R(U,V)W,Y))

= (h(XfZ)1Ri(U,V)h(W>Y))-{h(YfZ)fR

i(U}V)h(X>W)> -<h(X,W),Ri(U,V)b(Y,Z)) + <b(Y,W),Ri(U,V)h(X,Z))

= 0 . •

The proof of the following lemma is trivial.

LEMMA 2. Let ij : Mni —>M™* be isometric immersions (j E {1,. . . ,&}) and ii x ... x ik the product immersion. Then ix x ... x ik is semi-parallel if and only if *!,..., ik are semi-parallel. •

4. Reduction of the condition R 'h = 0.

Let i'-iAPfg)—>En+1 be an isometric immersion. Let xEMn. Choose a basis {e1,...,en} of TXM such that Ae^^ei ( i 'E{l, . . . ,»}). Then R(ej, £y) = X,- Xy e\ A £y and h (e^, ej) = X(- by. This implies that (5fa, */) -b) {ekied = \ Xy (X,- - Ay) (8jk 6,-g - 8ik 6y8), which gives that i is semi-parallel if and only if XjX^X,- — Xy) = 0 for all mutually distinct i , / E E {1 , . . . , n}. From this one obtains that an immersion is semi-parallel if and only if at each point, for a suitable choice of indices, one has

Aei = 0 (3)

for * E {1,. . . ,k} and

Aei = Xe,-

for iG{fe + l,... ,w} with &E{0, . . . ,»} and XEIR0. Notice that by (1) all sectional curvatures are positive or equal to zero.

Denote the multiplicity of the eigenvalue X by s\. As X is a continuous function on M (see [7]) one knows that if S\(p) = £ for p in M there exists a neighbourhood U of p in M such that s\(q) > £ for all q E [/.

Let Af' be a connected component of {# EM | sx(#) ^ 0}- M' is an open part of M. By proposition 2.2 in [7] one concludes that X is a differ­entiate function on Mf and s^ is a constant on M'. Let 7\(#) := {X E ET^M|i4X = XX} and T0(4) := {XET^Af | AY = 0} for 4 EAT. By

310

proposition 2.3 in [7] the distributions 7 \ and T0 are differentiable. From the equations of Codazzi (2) one obtains that

[X,Y]eT0,

(X'\)Z + AVzX = 0, (4) (Z-X)F-(K-X)Z = 0 , iz,v]erx.

For all vectorfields X, Y in TQ and Z, V in 7\ . This shows that T0 and 7 \ are involutive. If s\(q)>2, then X is a constant along the integral manifolds of T\ and the ^-decomposition of TqM is TqM - V^0)(q) 4-4- VWty) where ^<0)(^) = T0(q) and V^O?) = 7\(#). This implies that M1 is an infinitesimally irreducible simple leaf or Af' is locally isometric to M0 XMX where M0 has zero curvature, i.e. for every pEM' there exists a neighbourhood U of p, open sets l/0 of Af0 and C/j of Afj and an isometry \p : U0 x t/j —• £/. If nx := dimAfj, then «j ^ s x . Moreover, if X is tangent to U0 and y is tangent to Ulf then ib(^#(X), ^#(y)) = 0 because Z(0)(q)CT0(q) for every ^ EAtf'. By a lemma in [5] we conclude that the immersion i: M'—>En*1 is locally congruent to the product of immersions i0 : M0—>En ni and ix: Afj—>-l?Wl . f'o(Mo) *s a n °Pen submanifold of En ni and ix is semi-parallel by Lemma 2.

Choose a system of coordinates (xlt...,xn) on M' such that the integral manifolds of 7 \ are (locally) given by the equations

(x1=a1

j (5.1) \ xk=ak

and those of T0 by

i ^fe+i = ak+i

(5.2) Xn ~~ &n

((alt:..,an)eiR» and * : = w - s x )

and moreover such that

311

and

sdk'^k)=Sii (5-4) for j , / e { l , . . . , * } .

The generalized second fundamental forms (defined in [9]) £, and M\ (i,j E {1,. . . ,k}) are given by

V , ^ - = B , . W + . | i M | ( X ) ^ -

for X G 7 \ and B,(X) = 0, Mii(X) = 0 for X G r 0 . riX

From (4) we obtain that -r— X + X.B, (X) = 0, from which

Bi(X) = mX (6)

with

"':=i-(lnT) <7 )

for j G { l , ...,*} and XGT X .

5. Examples of semi-parallel hypersurfaces in £w + 1 .

Hyperplanes (i.e. totally geodesic immersions), hyperspheres (i.e. totally umbilical immersions) and cylindrical immersions are semi-parallel. Next we introduce another example.

Consider the set

C={(x0,xl,...,xn)ejRn+i\x0>0 and (tg20)*J = x\ + ... + x2n]

with 0 T£ 0. For the induced differentiable and geometrical structure the inclusion mapping i is an isometric immersion. A standard imbedding of an elliptic cone ([9]) is an immersion congruent to i. C is foliated by lines ^ . . ^ - { ( ^ ^ l ^ — ^ O l ^ I R j } where X2 + ... + X2 = tg20 and by

hyperspheres S\ : = {(\xlf... ,xn)\ xlf...,xn G 1R and S xj = X2 tg20}. i = i

Choosing a local orthonormal frame adapted to these foliations one obtains the following form for the second fundamental tensor in the point

312

A =

which shows by (3) that i is semi-parallel. It is easily seen that C is not complete.

Products of hyperspheres and cones with Euclidean spaces are semi-parallel as well.

6. Proof of Theorem 1.

First we consider the case s\ = n. Then M' is an infinitesimally irreduc­ible simple leaf. Then i: M' —• En+1 is a totally umbilical immersion and so it is an open part of a standard imbedded w-sphere Sn in En+1. Moreover, X is a constant on M\ which implies that M' is closed in M. As M is connected, M =M'.

Next we treat the case 2<s\<n. We examine the immersion

ix: Mnxl —>Eni+ . As s\ = dim V*1* > 2, we get by Lemma 4.5 from [9]

and Lemma 1 that s^ < nx < s ^ + 2.

If nx = s\, then ix is totally umbilical and so an open part of a stan­dard imbedded nl -sphere in Enx . Moreover, X is a constant on M' and one concludes in the same way as above that M* = M. So i is an immersion of type (iv) in the theorem.

If Si = s\ + 1, using Theorem 4.4 from [9], one concludes that Mx is locally isometric to an elliptic cone, a hyperbolic cone or a Euclidean cone (for definitions, see [9]). As the last two have negative curvature, one obtains that Mx is locally isometric to an elliptic cone. Using Theorem 1.3 from [7], it is seen that ix is locally congruent to the immersion of an elliptic cone discussed in section 5. From the form of A, it is clear that M' is closed in M and thus M' = M. This proves that i is an immersion of type (ii) or (v) in Theorem 1.

0 cos2 d 1 sin 6 x0

cos2fl 1 sin 0 x0

313

It is not possible that nt = s\ + 2. Indeed, it follows from Lemma 4.6 in [9] that in this case not all generalized second fundamental forms are sym­metric, which contradicts (6).

We examine the case s\ = 2. Choose a system of coordinates (xl,x2)

...,xn) and define functions na (aG {1 , . . . , n — 2}) as in section 4. From

(5) it follows that -r = 0 and -r— = 0 and so also -r—— = - a — 0 oxn-i oxn o#»-i oxn

for all a by (7). Formula (2.10) in [9] implies that

for all a, |3 G {1 , . . . , n - 2}. We solve this system of differential equations.

If Ma = 0 for some a G {1 , . . . , n - 2} then -=-**- = 0 for all 0. Restrict

the attention to a connected open set U on which Mi,... ,M£ never vanish and Mfi+1 = ... =M„-2 = 0 (£ G {0,...,« - 2}). If £ = 0, then Z(0) = V<0)

and so ij is an open part of a standard imbedding of a sphere. X is seen to be a constant on M' so that M' =M as above, i is an immersion of type (iv) in Theorem 1. We consider the case £ > 1.

Combining -~ \- fxafi^ = 0 and -r-^- 4- fXo = 0, one obtains that

,ln (£))="4r (ln t^))and so that Ma=K0lx&'with 4r *«*= bxp

= 0 for a ,0G{ l , . . . , £} . For ot,P,ye {1, . . . ,£}, one has

raa = 1 >

from which one obtains that all ^ a£ are constants which are moreover nonzero. We find that

/ * i = - l

2 ilffa-xp +t1

and so 1

Ma — g 2 ^/ja*0 + *«

314

with ta := tt )jj10l ( aG{ l , . . . ,£}). Put a : = with

P :

-tf 2 fy', and take PGO(w)

> 1 1 .-*«1

0 0

*</'"

0

0 'Jn-i -

Consider the coordinate transformation

yi X j

. = p

# +

• •

-y». . xn- . 0 _

Then /*« = . It is clear that the coordinate system (yx,..., yn) satisfies

the conditions (5). Let B'a be the generalized fundamental forms associated t o (yi , . . . , yn)

a n d M« the corresponding functions ( a G { l , . . . , » - 2 } ) . Then ti'2 = ... = / 4 - 2

= 0> a n ^

M i - 2 ^ M a = S * « •

a = i tf

We assert that Z(1)(p) = vect

a = i tf

d

J_

^ ( P ) ] + ^>(P) for each p in U.

From formula (2.8) in [9] one obtains that M% (Y)B'1 (X) -M^ (X)B\(Y) = 0 for X, Y with values in ]A0) and a € { 2 , . . . , » - 2}. This implies that

A41 = 0, i.e. < Vx tyx ' ^a > = 0 for all X with values in ^ 0 ) and all

a. G {2,. . . ,n - 2}. Together with < Vx F, -~— > = 0 for all X, Y with values

in P*0) and all a E { 2 , . . . , » - 2}, this proves the assertion. Q

One obtains that nl = 3 and by (7) that X = — for cEIR0 . It

follows that A is the second fundamental form of the standard imbedding of C3 in E4. In the same way as before it follows that U is closed in M' so that the immersion i is an immersion-of type (ii) or (v) in Theorem 1.

315

Finally, in case s A < l , the immersion is cylindrical. This proves Theorem 1. •

7. Proof of Theorem 3.

Let i : (Mn,g)—>En+1 be an isometric immersion, i is semi-parallel if and only if R(X, Y)A := [R(X, Y), A] = 0 for all tangent vectors X and Y. We study isometric immersions i for which C{X,Y)A = 0 for all X, Y in TM. For xGMn and a basis {el,...,en} for TXM as in section 4 one has the following expression for C:

where

C(ei,ej) = aijei/\ej ,

ajj :=X,X;- —(ki(trA-Xi) + Xy(tr A -Xy))

2 — 4-« A 2

4-(trAy-trA (n-l)(n-2)

(see f.i. [3]). It is easily seen that i satisfies the condition C'A = 0 if and only if

^y(X,-Xy) = 0 (8)

for all f,/€E {1,2,...,»}'. Let X^.^Xp be the (mutually) distinct eigen­values of A with multiplicities slf...,Sp respectively. Denote by Wa the space of eigenvectors with eigenvalue Xa. If £,, £feGWa and e-t e%€zVa for i=fcj and k=£9., then a^ — ak^. We define numbers bap = ajj, where i # ; , e,- G Wa and ^ G ^ (a, 0 G {1,...,/?}). The implication (ii) =• (i) in Theorem 3 holds trivially. To prove that (i) implies (ii), it is sufficient to show that bap = 0 for all a, 0 for which bap is defined. It is seen from (8) that bap = 0 for ot and j8 distinct. We show that this implies that i is quasiumbilical. Suppose that p>3. From bl2 = b23 = b31 — 0 one obtains that (n - 3)(XX — X3) = 0, which yields a contradiction. If p = 2, then

_ ( s 1 - l ) ( s 2 - l ) ( X 1 - X 2 ) 2

12 ( i t - 1 ) ( » . - 2 )

so that &12 = 0 implies that s t = 1 or s2 = 1. •

316

BIBLIOGRAPHY

[1]. E.Cartan, La deformation des hypersurfaces dans Vespace con forme, reel a n^5 dimensions, Bull. Soc. Math. France 45 (1977), p. 57-121.

[2] E. Cartan, Lecons sur lageometrie des espaces de Riemann, 2nd ed., Paris, 1946.

[3] J.Deprez, P. Verheyen and L. Verstxaelen, Characterizations of con formally flat hypersurfaces, to appear.

[4] D. Ferus, Symmetric Submanifolds of Euclidean Space, Math. Ann. 247, 81-93 (1980).

[5] J.D. Moore, Isometric immersions of Riemannian products, J. Differential Geo­metry 5 (1971), 159-168.

[6] K. Nomizu, On hypersurfaces satisfying a certain condition on the curvature tensor, Tohoku Math. J. 20 (1968) 46-59.

[7] P.J.Ryan, Homogeneity and some curvature conditions for hypersurfaces, T6hoku Math. J. 21 (1969), 363-388.

[8] P.J. Ryan, A class of complex hypersurfaces, Colloq. Math. 26 (1972), 175-182.

[9] Z.I. Szabo, Structure theorems on Riemannian spaces satisfying R(X,Y)'R = 0. I. The local version, J. Differential Geometry 17 (1982) 531-582.

J OH AN DEPREZ - Katholieke Universiteit Leuven - Departement Wiskunde Celestijnen-laan200B - B-3030 Leuven, Belgium.

Lavoro pervenuto in redazione l'l/X/1984