Hypersurfaces in space forms satisfying the condition
∆x = Ax + B
Luis J. Alı́as, Angel Ferrández and Pascual Lucas
Trans. Amer. Math. Soc. 347 (1995), 1793–1801
(Partially supported by DGICYT grant PB91-0705)
Abstract
In this work we study and classify pseudo-Riemannian hypersurfaces in pseudo-Riemannian
space forms which satisfy the condition ∆x = Ax + B, where A is an endomorphism, B
is a constant vector and x stands for the isometric immersion. We prove that the family
of such hypersurfaces consists of open pieces of minimal hypersurfaces, totally umbilical
hypersurfaces, products of two non-flat totally umbilical submanifolds and a special class of
quadratic hypersurfaces.
1. Introduction
Let x be an isometric immersion of a hypersurface Msn in Rn+1
and assume there exist an
t
n+1
n+1
endomorphism A of Rt and a constant vector B in Rt such that ∆x = Ax + B. We ask for
the following question: “What is the geometric meaning involved in that algebraic condition?”
This question was first studied in the Euclidean case by Chen and Petrovic [4], Dillen, Pas and
Verstraelen [5], and Hasanis and Vlachos [7], who obtained some interesting classification theorems. Recently, Park [10], following closely the ideas in [2] and [1], has considered that condition
with B = 0 for hypersurfaces in Euclidean spherical and hyperbolic spaces.
To study that question in its full generality, it seemed natural to us to begin with Lorentzian
surfaces, [2]. Later, in [1], in order to generalize the above papers we gave a classification theorem
for pseudo-Euclidean hypersurfaces. Actually, we proved that the only hypersurfaces satisfying
the matricial condition on the Laplacian are open pieces of minimal hypersurfaces, totally umbilical hypersurfaces and pseudo-Riemannian products of a totally umbilical and a totally geodesic
submanifold.
This paper arises as a natural continuation of [2] and [1], taking now a non-flat pseudoRiemannian space form as the ambient space. Here, we analyze the isometric immersions x of
a hypersurface Msn of M̄νn+1 satisfying ∆x = Ax + B, where M̄νn+1 is the pseudo-Euclidean
⊂ Rn+2
or the pseudo-Euclidean hyperbolic space Hn+1
⊂ Rn+2
sphere Sn+1
ν
ν
ν
ν+1 .
In this new situation, the codimension of the manifold Msn in the pseudo-Euclidean space
where it is lying is two, so that one hopes to find a richer family of examples satisfying the asked
condition. On the other hand, although the proofs given in [1] do not work here, we follow the
techniques developed there.
Before to refer to the main result, we wish pointing out that a lot of hypersurfaces having nondiagonalizable shape operator are given. This property makes substantially the difference between
this case and that treated in [1].
The main result of this paper states that the only hypersurfaces Msn of M̄νn+1 satisfying the
matricial condition on the Laplacian are open pieces of minimal hypersurfaces, totally umbilical
1
Trans. Amer. Math. Soc. 347 (1995), 1793–1801
hypersurfaces, pseudo-Riemannian products of two non-flat totally umbilical submanifolds and
quadratic hypersurfaces defined by {x ∈ Rn+2
: hx, xi = ±1, hLx, xi = c}, where L is a
t
n+2
self-adjoint endomorphism of Rt
with minimal polynomial µL of degree two, and c is a real
constant such that µL (kc) 6= 0.
2. Preliminaries
Let Rn+2
be the (n + 2)-dimensional pseudo-Euclidean space whose metric tensor is given by
t
2
ds = −
t
X
i
i
dx ⊗ dx +
i=1
n+2
X
dxj ⊗ dxj ,
j=t+1
where (x1 , . . . , xn+2 ) is the standard coordinate system. For each k 6= 0, let M̄νn+1 (k) be the
complete and simply connected space with constant sectional curvature sign(k)/k 2 . A model for
M̄νn+1 (k) is the pseudo-Euclidean sphere Sn+1
(k) if k > 0 and the pseudo-Euclidean hyperbolic
ν
n+1
n+1
space Hν (k) if k < 0, where Sν (k) = {x ∈ Rn+2
: hx, xi = k 2 } and Hn+1
(k) = {x ∈
ν
ν
n+2
2
Rν+1 : hx, xi = −k }, h, i standing for the indefinite inner product in the pseudo-Euclidean
space. Throughout this paper we will assume, without loss of generality, that k 2 = 1.
¯ and ∇)
˜ denotes the
Let Msn be a pseudo-Riemannian hypersurface in M̄νn+1 and let ∇ (∇
n+2
n
n+1
Levi-Civita connection on Ms (M̄ν
and Rt , respectively). We will also denote by N the
n
n+1
unit normal vector field to Ms in M̄ν . Let H 0 and H be the mean curvature vector fields of
Msn in M̄νn+1 and Rn+2
, respectively. Thus we may write H 0 = αN , α being the mean curvature
t
n
n+1
of Ms in M̄ν , and
H = H 0 − kx = αN − kx.
Let x : Msn −→ M̄νn+1 be an isometric immersion satisfying the condition
∆x = Ax + B,
where A is an endomorphism of Rn+2
and B a constant vector in Rn+2
. Taking covariant derivat
t
tive in (2) and using the Laplace-Beltrami equation ∆x = −nH and the Weingarten formula we
get AX = nSH X − nDX H, for any vector field X tangent to Msn , where D denotes the normal connection on Msn and Sξ the Weingarten endomorphism associated to a normal vector field
ξ. Then by (1) we have DX H = X(α)N and SH X = αSX + kX, where, for short, we have
written S for the Weingarten endomorphism SN . From now on, we will call S the shape operator
of Msn . Now from the above formulae we deduce that
AX = n(αSX + kX) − nX(α)N.
From (2), taking into account the Laplace-Beltrami equation and (1), we obtain the following
equation
Ax = −nαN + nkx − B.
By applying the Laplacian on both sides of (2) and using again that ∆x = −nH we find AH =
∆H, that along with (1) leads to αAN = ∆H + kAx. Now, bringing here (4) and the formula
for ∆H obtained in [3, Lemma 3]
∆H = 2S(∇α) + nεα∇α + (∆α + εα|S|2 + nkα)N − nk(k + εα2 )x,
2
Luis J. Alı́as, Angel Ferrández and Pascual Lucas, Hypersurfaces in space forms satisfying the condition ∆x = Ax + B
where ∇α stands for the gradient of α, ε = hN, N i and |S|2 = trace(S 2 ), we get the following
equation
αAN
= 2S(∇α) + nεα∇α + (∆α + εα|S|2 )N
(5)
2
− nkεα x − kB.
3. Some examples
In this paper we wish to classify the hypersurfaces Msn in M̄νn+1 whose isometric immersion
satisfies the condition (2). In order to get such a classification we need some examples.
2.1 Minimal hypersurfaces Msn in M̄νn+1 obviously satisfy (2). Indeed, by using (1) we have
H = −kx and ∆x = nkx. So we can take A = nkIn+2 and B = 0.
2.2 Let Msn be a totally umbilical hypersurface in M̄νn+1 . Taking into account the classification
theorem for such hypersurfaces (see, for example, [9, Theorem 1.4]) we get, according to hH, Hi
is positive, negative or zero, Msn is an open piece of a pseudo-Euclidean sphere Sns (r), a pseudoEuclidean hyperbolic space Hns (r) or Rns . In the last case, the immersion f : Rns −→ Rn+2
s+1
is given by f (u) = (q(u), u1 , . . . , un , q(u)), where q(u) = a hu, ui + hb, ui + c, a 6= 0. The
pseudo-Euclidean spheres and pseudo-Euclidean hyperbolic spaces both satisfy the condition (2).
Indeed, by considering ϕ as the standard immersion of Sns (r) or Hns (r) in a hyperplane Rn+1
s0
n+1
of Rn+2
,
we
know
from
[1]
that
∆ϕ
=
Lϕ,
L
being
an
endomorphism
of
R
.
The
(n
+
t
s0
1) × (n + 1) matrix L and the immersion ϕ become an (n + 2) × (n + 2) matrix A (filling
with zeros) and an immersion x in Rn+2
, respectively, in a natural way and so we get (2) with
t
B = 0. Therefore the most interesting case is that with hH, Hi = 0. Now we can choose a
n
point p in Rn+2
s+1 such that hf − p, f − pi = ±1 and then x = f − p is an immersion from Rs
n+1
in M̄ν
with ∆x = −2n(a, 0, . . . , 0, a). Thus this hypersurface satisfies (2) with A = 0 and
B = (−2na, 0, . . . , 0, −2na). Furthermore, from the equation ∆x = −nαN + nkx, we easily
obtain that its constant mean curvature α is given by α2 = 1.
0
m0 +1
2.3 Let x : Msm −→ Rm+1
and y : M 0m
be two isometric immersions satisfying
t
s0 −→ Rt0
the condition (2) and let z = x×y be the natural isometric immersion from the pseudo-Riemannian
0
m+m0 +2
product Msm × M 0m
. If ∆x = Ax + B and ∆0 y = A0 y + B 0 then we can consider
s0 in Rt+t0
˜ = Ãz + B̃. Then from
à = diag[A, A0 ] and B̃ = (B, B 0 ). Thus it is easy to show that ∆z
[1, Section 2], we can construct the following examples of hypersurfaces in M̄νn+1 satisfying the
condition (2):
p
√
n+1 , with 0 < r < 1 and r 6=
2
p/n, whose constant mean
(a) Spu (r) × Sn−p
s−u ( 1 − r ) ⊂ Ss
2
2
2
2
2
2
curvature is given by α √
= (nr − p) /(n r (1 − r ));
p
(b) Hpu (−r) × Hn−p
1 − r2 ) ⊂ Hn+1
p/n and α2 = (nr2 −
(−
s−u
s+1 , with 0 < r < 1, r 6=
p)2 /(n2 r2 (1 − r2 )); √
n+1
2
2
2
2 2 2
2
(c) Spu (r) × Hn−p
s−u (− √−1 + r ) ⊂ Ss+1 , with r > 1 and α = (nr − p) /(n r (r − 1));
p
n−p
, with r > 0 and α2 = (nr2 + p)2 /(n2 r2 (1 + r2 ));
(d) Su (r) × Hs−u (− 1 + r2 ) ⊂ Hn+1
s
where 1 6 p 6 n − 1 and 0 6 u 6 s. We will refer these examples as the pseudo-Riemannian
non-minimal standard products.
2.4 The hypersurfaces in examples 2.2 and 2.3 have diagonalizable shape operator. However,
it seems natural thinking of hypersurfaces with non-diagonalizable shape operator satisfying (2)
into indefinite ambient spaces. Let L be a self-adjoint endomorphism of Rn+2
, that is, hLx, yi =
t
n+1 −→ R be the quadratic function defined by f (x) =
hx, Lyi for all x, y ∈ Rn+2
.
Let
f
:
M̄
ν
t
3
Trans. Amer. Math. Soc. 347 (1995), 1793–1801
hLx, xi and assume that the minimal polynomial of L is given by µL (t) = t2 + at + b, a, b ∈ R.
˜ (x) = 2Lx and ∇f
¯ (x) =
Then by computing the gradients, at each point x ∈ M̄νn+1 , we have ∇f
n+2
n+1
˜
¯
2Lx − 2kf (x)x. If ∆ and ∆ denote the Laplacian operators on Rt and M̄ν , respectively, a
˜ (x) = −2 trace(L) and ∆f
¯ (x) = −2 trace(L) − 2k(n +
straightforward computation yields ∆f
1)f (x).
Consider the level set M = f −1 (c) for a real constant c. Then at a point x in M we have
®
­
®
­
¯ (x), ∇f
¯ (x) = 4 L2 x, x − 4kf (x)2 = −4kµL (kc),
∇f
and so f is an isoparametric function (see [6]). Thus the level hypersurfaces {f −1 (c)}c∈I , where
I ⊂ {c ∈ R : µL (kc) 6= 0} is connected, form an isoparametric family in the classical sense.
1 ¯ ¯
1
The shape operator of Msn is given by SX = − ¯ ∇
(kcX − LX) and
X (∇f ) =
|∇f |
|µL (kc)|1/2
a messy computation gives
nkc − tr(L) − a
tr(S) =
.
|µL (kc)|1/2
Then the mean curvature α is given by
α=
ε
a + tr(L) − nkc
tr(S) = δ
,
n
nk|µL (kc)|1/2
where δ stands for the sign of µL (kc). Therefore we get
H0 =
a + trL − knc
(Lx − kcx),
knµL (kc)
from which we deduce, by using ∆x = −n(H 0 − kx), that ∆x = Ax, where A is given by
A=
knc − a − trL
ctrL + (n + 1)ac + knb
L+
In+2 .
kµL (kc)
µL (kc)
4. First characterization results
The aim of this section is to show that a hypersurface Msn of M̄νn+1 satisfying the condition
(2) has to be of constant mean curvature. To do that, let W be the open set of regular points of α2 ,
which we may assume a non-empty set. From (3) we have hAX, xi = 0, for any vector field X
tangent to Msn . Taking covariant derivative there we get
hAσ(X, Y ), xi = − hAX, Y i ,
,
for all tangent vectors X and Y , where σ represents the second fundamental form of Msn in Rn+2
t
which is given by
σ(X, Y ) = ε hSX, Y i N − k hX, Y i x.
Now equation (1), jointly with (3) and (2), leads to
εhSX, Y ihAN, xi − khX, Y ihAx, xi = −nαhSX, Y i − nkhX, Y i.
Bringing here the formulae for Ax and AN given in (4) and (5), respectively, a straightforward
computation yields
hSX − εαX, Y i hB, xi = 0,
at the points of W. This equation is the key to the following result.
4
Luis J. Alı́as, Angel Ferrández and Pascual Lucas, Hypersurfaces in space forms satisfying the condition ∆x = Ax + B
Lemma 4.1 Let x : Msn −→ M̄νn+1 be a hypersurface such that ∆x = Ax + B. If Msn has
non-constant mean curvature, then B = 0.
Proof. Let us consider the set U = {p ∈ W : hB, xi (p) 6= 0} and assume it is a non-empty set.
Then at the points of U, from (3) and (3), we have
AX = n(εα2 + k)X − nX(α)N.
Since n > 2, we can always find a vector field X such that X(α) = hX, ∇αi = 0. This shows,
by using (4), that n(εα2 + k) is an eigenvalue of A and therefore locally constant on U, which is
a contradiction. Hence U = ∅ and hB, xi = 0 on W. Taking covariant derivative here he deduce
that B has no tangent component and therefore we get B = ε hB, N i N and hB, N i = 0, because
W is not empty.
Next we are going to make some computations before to state the main result of this section.
From equation (3) it is easy to see that
hAX, Y i = hX, AY i ,
for all tangent vector fields X and Y . Taking covariant derivative here and using the Gauss formula
jointly with (5), we find
hAσ(X, Z), Y i − hAσ(Y, Z), Xi =
(6)
hσ(X, Z), AY i − hσ(Y, Z), AXi .
By (2) and (3), the equation (6) becomes
εhSX, ZihAN, Y i − khX, ZihAx, Y i −
(7)
εhSY, ZihAN, Xi + khY, ZihAx, Xi =
−nY (α)hSX, Zi + nX(α)hSY, Zi.
Finally, by Lemma 4.1, (4) and (5), from (7) we obtain
T X(α)SY = T Y (α)SX,
(8)
where T means the self-adjoint operator defined by T X = nαX + εSX. This equation becomes
the crucial point to show the next result.
Proposition 4.2 Let x : Msn −→ M̄νn+1 be an isometric immersion such that ∆x = Ax + B.
Then Msn has constant mean curvature.
Proof. From Lemma 4.1 we can assume B = 0 and then equation (8) holds on W. First, suppose
that T (∇α) 6= 0 at the points of W. Then there is a vector field X tangent to Msn such that
T X(α) 6= 0, so that by using (8) we find that rankS=1 at the points of W. Therefore we can
choose a local orthonormal frame {E1 , . . . , En } such that SE1 = nεαE1 , SEi = 0, i = 2, . . . , n
and εi = hEi , Ei i. Also from (8) we have that Ei (α) = 0, i = 2, . . . , n and using again (3), (4)
and (5) we get
AE1 = n(k + εnα2 )E1 − nE1 (α)N,
AEi = nkEi , i = 2, . . . , n,
∆α
AN = 3nεε1 E1 (α)E1 + {
+ εn2 α2 }N − nkεαx,
α
Ax = −nαN + nkx.
5
Trans. Amer. Math. Soc. 347 (1995), 1793–1801
Therefore, span{E1 , N, x} is an invariant subspace under A and the characteristic polynomial
pA (t) of A is given by pA (t) = (t − nk)n−1 pA∗ (t), where A∗ stands for A|span{E1 ,N,x} . Then
pA∗ (t) is constant and we can find three real constants λ1 , λ2 and λ3 (which are nothing but the
invariants associated to A∗ ) such that
∆α
+ 2n{k + εnα2 },
α
∆α
= n(2k + εnα2 )(
+ εn2 α2 ) + 3n2 εε1 E1 (α)2 + n2 k(k + εnα2 ) − kn2 εα2 ,
α
∆α
+ εn2 α2 ) − n3 kεα2 (k + εnα2 ) + 3n3 εε1 kE1 (α)2 .
= n2 k(k + εnα2 )(
α
λ1 =
λ2
λ3
Then we obtain
nkλ2 = λ3 + n3 (k + εnα2 ) + n2 (
and
∆α
+ εn2 α2 ) + n4 kα4
α
∆α
= λ1 − 2n(k + εnα2 ).
α
Last two equations allow us to write n4 α4 = kpA∗ (kn) and so α is locally constant on W, which
is a contradiction.
Finally, assume now there is a point p in W such that T (∇α)(p) = 0. Note that from (3) and
(5) we have in W, ∀i = 1, . . . , n
hAEi , N i = −nεEi (α)
2ε
hEi , AN i =
hT (∇α), Ei i − nεEi (α).
α
(9)
hAEi , N i = hEi , AN i.
(10)
It follows that at p,
From (3), (4), (5), (1) and (10) we deduce that A is a self-adjoint endomorphism of Rn+2
and thus
t
equation (10) remains valid at every point in W. In turn, from (10) T (∇α) = 0 on W and so
∇α is an eigenvector of S with associated eigenvalue −nεα. If h∇α, ∇αi = ∇α(α) = 0, from
(3) we could write A(∇α) = n(k − nεα2 )∇α, then n(k − nεα2 ) should be an eigenvalue of A
and α must be locally constant on W, which cannot be hold. Therefore we can choose a local
orthonormal frame {E1 , . . . , En } with E1 parallel to ∇α such that
AE1 = n(k − nεα2 )E1 − nE1 (α)N,
AEi = n(αSEi + kEi ),
i = 2, . . . , n,
∆α
AN = −nεε1 E1 (α)E1 + {
+ ε|S|2 }N − nkεαx,
α
Ax = −nαN + nkx.
Since SEi ∈ span{E2 , . . . , En }, i = 2, . . . , n, we may write S ∗ for the endomorphism S restricted to span{E2 , . . . , En }. Working on the characteristic polynomials, from the above equations we can deduce that
t − nk
pA (t) = (nα)n−1 q(t)pS ∗ (
),
nα
6
Luis J. Alı́as, Angel Ferrández and Pascual Lucas, Hypersurfaces in space forms satisfying the condition ∆x = Ax + B
where q(t) is a polynomial of degree three. Let {r1 , . . . , rn } be the possibly complex roots of
pS (t), with r1 = −nεα and {r2 , . . . , rn } the roots of pS ∗ (t). Then the functions nk + nαrj , j =
2,
P.n. . , n, are roots of pA (t) and therefore
Pn they are locally constant on W. Thus from the formula
2
(nk+nαr
)
=
n(n−1)k+nα
j
j=2
j=2 rj = n(n−1)k+nα(trS −r1 ) = n(n−1)k+2εn α,
we obtain that α is locally constant on W, which is a contradiction.
Summarizing, we have got that W has to be empty, i.e., Msn has constant mean curvature.
5. Main results
We have just proved that the hypersurfaces Msn of M̄νn+1 satisfying the condition (2) have
constant mean curvature. In this section, we wish to give a classification theorem of such a class
of hypersurfaces. To do that, we recall the following definition. A hypersurface Msn is said to
be isoparametric if the characteristic polynomial pS (t) of its shape operator S is the same at all
points of Msn . When S is diagonalizable (for example, in the definite case) that means that the
principal curvatures of Msn , as well as its multiplicities, are constant. Our first main result states
as follows.
Theorem 5.1 Let x : Msn −→ M̄νn+1 be an isometric immersion satisfying ∆x = Ax + B. Then
Msn is a minimal or an isoparametric hypersurface.
Proof. Let Msn be a hypersurface of M̄νn+1 satisfying (2). By Proposition 4.2 we can assume that
the mean curvature α of Msn in M̄νn+1 is a non-zero constant and so the equation (3) works here.
If Msn is not totally umbilical in M̄νn+1 , then we have hB, xi = 0 and, as in Lemma 4.1, B = 0.
Now, from (3), (4) and (5) we get
AX = n(αSX + kX),
AN
= ε|S|2 N − nkεαx,
(1)
Ax = −nαN + nkx.
Working again on the characteristic polynomials pA (t) and pS (t), as in Section 4, we deduce that
pS (t) is constant on Msn and the proof finishes.
Next with the aim of getting a complete classification of those hypersurfaces Msn of M̄νn+1
satisfying (2), some easy computations are needed. From Theorem 5.1, Msn is an isoparametric hypersurface provided that the (constant) mean curvature α is not zero, and thus |S|2 is also
˜ X (AN ) =
constant. Taking covariant derivative in the expression of AN in (1) we have ∇
2
2
˜
˜
−ε|S| SX − nkεαX and ∇X (AN ) = A(∇X N ) = −n(αS X + kSX), from which we obtain
S2 +
nk − ε|S|2
S − kεI = 0,
nα
(2)
where I stands for the identity operator on the tangent bundle of Msn . We have just seen that if
Msn is not totally umbilical then B = 0 and thus equations (1) and (2) allow us to write
A2 − (ε|S|2 + kn)A + nεk(|S|2 − nα2 )In+2 = 0,
and furthermore, from (1), A is a self-adjoint endomorphism of Rn+2
. If S is diagonalizable,
t
from (2), Msn has exactly two constant principal curvatures. By using now similar arguments as
7
Trans. Amer. Math. Soc. 347 (1995), 1793–1801
in Theorem 2.5 of [11] and Lemma 2 of [8], we deduce that Msn is an open piece of a pseudoRiemannian product of two non-flat totally umbilical submanifolds. If S is not diagonalizable,
from (3), the minimal polynomial µA (t) of A is given by µA (t) = t2 + at + b, with a = −(ε|S|2 +
kn) and b = nεk(|S|2 − nα2 ). Since hAx, xi = n is constant on Msn and µA (kn) = −n2 α2 εk 6=
0, then Msn is an open piece of a quadratic hypersurface as in example 2.4. Summing up, we have
proved the following theorem.
Theorem 5.2 Let x : Msn −→ M̄νn+1 be an isometric immersion. Then ∆x = Ax + B if and
only if Msn is an open piece of one of the following hypersurfaces in M̄νn+1 :
1) a minimal hypersurface,
2) a totally umbilical hypersurface,
3) a pseudo-Riemannian non-minimal standard product,
4) a quadratic hypersurface as in example 2.4, with non-diagonalizable shape operator (a2 −
4b 6 0).
As a consequence, we obtain the classification theorem for hypersurfaces in Sn+1 and Hn+1 ,
which generalizes Theorem 1.3 in [10].
Corollary 5.3 Let x : M n −→ M̄ n+1 be a non-minimal hypersurface. Then ∆x = Ax + B if
and only if M n is an open piece of one of the following hypersurfaces:
1) a totally umbilical hypersurface,
2) a product M̄ p (r1 ) × Sn−p (r2 ).
ACKNOWLEDGMENT
The authors would like to thank the referee for reading the paper carefully, providing some suggestions and comments to improve it.
Bibliography
[1]
L. J. Alı́as, A. Ferrández and P. Lucas. Submanifolds in pseudo-Euclidean spaces satisfying the
condition ∆x = Ax + B. Geom. Dedicata, 42 (1992), 345–354.
[2]
L. J. Alı́as, A. Ferrández and P. Lucas. Surfaces in the 3-dimensional Lorentz-Minkowski space
satisfying ∆x = Ax + B. Pacific J. Math., 156 (1992), 201–208.
[3]
B. Y. Chen. Finite-type pseudo-Riemannian submanifolds. Tamkang J. of Math., 17 (1986), 137–151.
[4]
B. Y. Chen and M. Petrovic. On spectral decomposition of immersions of finite type. Bull. Austral.
Math. Soc., 44 (1991), 117–129.
[5]
F. Dillen, J. Pas and L. Verstraelen. On surfaces of finite type in euclidean 3-space. Kodai Math. J.,
13 (1990), 10–21.
[6]
J. Hahn. Isoparametric hypersurfaces in the pseudo-riemannian space forms. Math. Z., 187 (1984),
195–208.
[7]
T. Hasanis and T. Vlachos. Hypersurfaces of En+1 satisfying ∆x = Ax + B. J. Austral. Math. Soc.,
Series A, 53 (1992), 377–384.
[8]
H. Lawson. Local rigidity theorems for minimal hypersurfaces. Ann. of Math., 89 (1969), 187–197.
8
Luis J. Alı́as, Angel Ferrández and Pascual Lucas, Hypersurfaces in space forms satisfying the condition ∆x = Ax + B
[9]
M. A. Magid. Isometric immersions of Lorentz space with parallel second fundamental forms.
Tsukuba J. Math., 8 (1984), 31–54.
[10] J. Park. Hypersurfaces satisfying the equation ∆x = Rx + B. To appear in Proc. A.M.S.
[11] P. J. Ryan. Homogeneity and some curvature conditions for hypersurfaces. Tôhoku Math. J., 21
(1969), 363–388.
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