E extracta mathematicae Vol. 32, Núm. 1, 55 – 81 (2017)

Characterizations of Complete Linear Weingarten
Spacelike Submanifolds in a Locally Symmetric

Semi-Riemannian Manifold

Jogli G. Araújo, Henrique F. de Lima,
Fábio R. dos Santos, Marco Antonio L. Velásquez

Departamento de Matemática, Universidade Federal de Campina Grande,
58.429 − 970 Campina Grande, Paráıba, Brazil

jogli@mat.ufcg.edu.br henrique@mat.ufcg.edu.br

fabio@mat.ufcg.edu.br marco.velasquez@mat.ufcg.edu.br

Presented by Manuel de León Received November 23, 2016

Abstract: In this paper, we deal with n-dimensional complete spacelike submanifolds Mn

with flat normal bundle and parallel normalized mean curvature vector immersed in an
(n + p)-dimensional locally symmetric semi-Riemannian manifold Ln+pp of index p obeying
some standard curvature conditions which are naturally satisfied when the ambient space is
a semi-Riemannian space form. In this setting, we establish sufficient conditions to guaran-
tee that, in fact, p = 1 and Mn is isometric to an isoparametric hypersurface of Ln+11 having
two distinct principal curvatures, one of which is simple.

Key words: Locally symmetric semi-Riemannian manifold, complete linear Weingarten
spacelike submanifolds, isoparametric submanifolds.

AMS Subject Class. (2010): 53C42, 53A10, 53C20, 53C50.

1. Introduction

Let L
n+p
p be an (n + p)-dimensional semi-Riemannian space, that is, a

semi-Riemannian manifold of index p. An n-dimensional submanifold Mn im-
mersed in L

n+p
p is said to be spacelike if the metric on M

n induced from that
of L

n+p
p is positive definite. Spacelike submanifolds with parallel normalized

mean curvature vector field (that is, the mean curvature function is positive
and that the corresponding normalized mean curvature vector field is parallel
as a section of the normal bundle) immersed in semi-Riemannian manifolds
have been deeply studied for several authors (see, for example, [2, 3, 15, 19]).
More recently, in [12] the second, third and fourth authors showed that com-
plete linear Weingarten spacelike submanifolds must be isometric to certain
hyperbolic cylinders of a semi-Riemannian space form Qn+pp (c) of constant
sectional curvature c, under suitable constraints on the values of the mean

55



56 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

curvature and of the norm of the traceless part of the second fundamental
form. We recall that a spacelike submanifold is said to be linear Weingarten
when its mean and normalized scalar curvature functions are linearly related.

Now, let L
n+p
p be a locally symmetric semi-Riemannian space, that is, the

curvature tensor R̄ of L
n+p
p is parallel in the sense that ∇R̄ = 0, where ∇ de-

notes the Levi-Civita connection of L
n+p
p . In 1984, Nishikawa [16] introduced

an important class of locally symmetric Lorentz spaces satisfying certain cur-
vature constraints. In this setting, he extended the classical results of Calabi
[4] and Cheng-Yau [6] showing that the only complete maximal spacelike hy-
persurface immersed in such a locally symmetric space having nonnegative
sectional curvature are the totally geodesic ones. This seminal Nishikawa’s
paper induced the appearing of several works approaching the problem of
characterizing complete spacelike hypersurfaces immersed in such a locally
symmetric space (see, for instance, [1, 10, 11, 13, 14]).

Our purpose in this paper is establish characterization results concerning
complete linear Weingarten submanifolds immersed in a locally symmetric
manifold obeying certain curvature conditions which extend those ones due to
Nishikawa [16]. For this, we need to work with a Cheng-Yau modified opera-
tor L and we establish a generalized maximum principle. Afterwards, under
suitable constrains, we apply our Omori-Yau maximum principle to prove that
such a submanifold must be isometric to an isoparametric hypersurface with
two distinct principal curvatures, one of them being simple. Our purpose in
this work is to extend the results of [10] for the case that the ambient space is a
locally symmetric semi-Riemannian manifold L

n+p
p obeying certain geometric

constraints. For this, in Section 3 we develop a suitable Simons type formula
concerning spacelike submanifolds immersed in L

n+p
p and having certain posi-

tive curvature function. Afterwards, in Section 4 we prove an extension of the
generalized maximum principle of Omori [17] to a Cheng Yau modified opera-
tor L (see Lemma 3). Moreover, we use our Simons type formula to obtain an
appropriated lower estimate to the operator L acting on the mean curvature
function of a linear Weingarten spacelike submanifold (cf. Proposition 1) and,
next, we establish our characterization theorems (see Theorems 1 and 2).



linear weingarten spacelike submanifolds 57

2. Preliminaries

Let Mn be a spacelike submanifold immersed in a locally symmetric semi-
Riemannian space L

n+p
p . In this context, we choose a local field of semi-

Riemannian orthonormal frames e1, . . . , en+p in L
n+p
p , with dual coframes

ω1, . . . , ωn+p, such that, at each point of M
n, e1, . . . , en are tangent to M

n.
We will use the following convention of indices

1 ≤ A, B, C, . . . ≤ n+p, 1 ≤ i, j, k, . . . ≤ n and n+1 ≤ α, β, γ, . . . ≤ n+p.

In this setting, the semi-Riemannian metric of L
n+p
p is given by

ds2 =
∑
A

ϵA ω
2
A,

where ϵi = 1 and ϵα = −1, 1 ≤ i ≤ n, n + 1 ≤ α ≤ n + p. Denoting by {ωAB}
the connection forms of L

n+p
p , we have that the structure equations of L

n+p
p

are given by:

dωA = −
∑
B

ϵB ωAB ∧ ωB, ωAB + ωBA = 0, (2.1)

dωAB = −
∑
C

ϵC ωAC ∧ ωCB −
1

2

∑
C,D

ϵCϵDRABCD ωC ∧ ωD, (2.2)

where, RABCD, RCD and R denote respectively the Riemannian curvature
tensor, the Ricci tensor and the scalar curvature of the Lorentz space L

n+p
p .

In this setting, we have

RCD =
∑
B

εBRCBDB, R =
∑
A

εARAA. (2.3)

Moreover, the components RABCD;E of the covariant derivative of the Rie-
mannian curvature tensor L

n+p
p are defined by∑

E

εERABCD;EωE = dRABCD −
∑
E

εE
(
REBCDωEA + RAECDωEB

+RABEDωEC + RABCEωED
)
.

Next, we restrict all the tensors to Mn. First of all,

ωα = 0, n + 1 ≤ α ≤ n + p.



58 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

Consequently, the Riemannian metric of Mn is written as ds2 =
∑

i ω
2
i . Since

−
∑
i

ωαi ∧ ωi = dωα = 0,

from Cartan’s Lemma we can write

ωαi =
∑
j

hαijωj, h
α
ij = h

α
ji. (2.4)

This gives the second fundamental form of Mn, B =
∑

α,i,j h
α
ijωi ⊗ ωjeα, and

its square length from second fundamental form is S = |B|2 =
∑

α,i,j(h
α
ij)

2.
Furthermore, we define the mean curvature vector field H and the mean cur-
vature function H of Mn respectively by

H =
1

n

∑
α

(∑
i

hαii

)
eα and H = |H| =

1

n

√√√√∑
α

(∑
i

hαii

)2
.

The structure equations of Mn are given by

dωi = −
∑
j

ωij ∧ ωj, ωij + ωji = 0,

dωij = −
∑
k

ωik ∧ ωkj −
1

2

∑
k,l

Rijklωk ∧ ωl,

where Rijkl are the components of the curvature tensor of M
n. Using the

previous structure equations, we obtain Gauss equation

Rijkl = Rijkl −
∑
β

(
h
β
ikh

β
jl − h

β
ilh

β
jk

)
. (2.5)

and

n(n − 1)R =
∑
i,j

Rijij − n2H2 + S. (2.6)

We also state the structure equations of the normal bundle of Mn

dωα = −
∑
β

ωαβ ∧ ωβ, ωαβ + ωβα = 0,



linear weingarten spacelike submanifolds 59

dωαβ = −
∑
γ

ωαγ ∧ ωγβ −
1

2

∑
k,l

Rαβklωk ∧ ωl.

We suppose that Mn has flat normal bundle, that is, R⊥ = 0 (equivalently
Rαβjk = 0), then Rαβjk satisfy Ricci equation

Rαβij =
∑
k

(
hαikh

β
kj − h

α
kjh

β
ik

)
. (2.7)

The components hαijk of the covariant derivative ∇B satisfy∑
k

hαijkωk = dh
α
ij −

∑
k

hαikωkj −
∑
k

hαjkωki −
∑
β

h
β
ijωβα. (2.8)

In this setting, from (2.4) and (2.8) we get Codazzi equation

Rαijk = h
α
ijk − h

α
ikj. (2.9)

The first and the second covariant derivatives of hαij are denoted by h
α
ijk and

hαijkl, respectively, which satisfy∑
l

hαijklωl = dh
α
ijk −

∑
l

hαljkωli −
∑
l

hαilkωlj −
∑
l

hαijlωlk −
∑
β

h
β
ijkωβα.

Thus, taking the exterior derivative in (2.8), we obtain the following Ricci
identity

hαijkl − h
α
ijlk = −

∑
m

hαimRmjkl −
∑
m

hαmjRmikl. (2.10)

Restricting the covariant derivative RABCD;E of RABCD on M
n, then Rαijk;l

is given by

Rαijkl = Rαijk;l +
∑
β

Rαβjkh
β
il +

∑
β

Rαiβkh
β
jl +

∑
β

Rαijβh
β
kl

+
∑
m,k

Rmijkh
α
lm.

(2.11)

where Rαijkl denotes the covariant derivative of Rαijk as a tensor on M
n.



60 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

3. Locally symmetric spaces and some auxiliary results

Proceeding with the context of the previous section, along this work we
will assume that there exist constants c1, c2 and c3 such that the sectional
curvature K and the curvature tensor R of the ambient space L

n+p
p satisfies

the following constraints:

K(u, η) =
c1
n
, (3.1)

for any u ∈ TM and η ∈ TM⊥; when p > 1, suppose that⟨
R(ξ, u)η, u

⟩
= 0, (3.2)

for u ∈ TM and ξ, η ∈ TM⊥, with ⟨ξ, η⟩ = 0. Suppose also

K(u, v) ≥ c2, (3.3)

for any u, v ∈ TM; and

K(η, ξ) =
c3
p
, (3.4)

for any η, ξ ∈ TM⊥.
The curvature conditions (3.1) and (3.3), are natural extensions for higher

codimension of conditions assumed by Nishikawa [16] in context of hyper-
surfaces. Obviously, when the ambient manifold L

n+p
p has constant sectional

curvature c, then it satisfies conditions (3.1), (3.2), (3.3) and (3.4). On the
other hand, the next example gives us a situation where the curvature condi-
tions (3.1), (3.2), (3.3) and (3.4) are satisfied but the ambient space is not a
space form.

Example 1. Let L
n+p
p = R

n1+p
p × Nn2κ be a semi-Riemannian manifold,

where Rn1+pp stands for the (n1+p)-dimensional semi-Euclidean space of index
p and Nn2κ is a n2-dimensional Riemannian manifold of constant sectional
curvature κ. We consider the spacelike submanifold Mn = Γn1 ×Nn2κ of L

n+p
p ,

where Γn1 is a spacelike submanifold of Rn1+pp .
Taking into account that the normal bundle of Γn1 ↪→ Rn1+pp is equipped

with p linearly independent timelike vector fields ξ1, ξ2, . . . , ξp, it is not diffi-
cult to verify that the sectional curvature K of L

n+p
p satisfies

K (ξi, X) =
⟨
RRn1+pp

(ξi, X1)ξ
i, X1

⟩
Rn1+pp

+
⟨
RNn2κ (0, X2)0, X2

⟩
N

n2
κ

= 0,
(3.5)



linear weingarten spacelike submanifolds 61

for each i ∈ {1, . . . , p}, where RRn1+pp and RN
n2
κ

denote the curvature tensors

of Rn1+pp and Nn2κ , respectively, ξi = (ξ
i, 0) ∈ T ⊥M and X = (X1, X2) ∈ TM

with ⟨ξi, ξi⟩ = ⟨X, X⟩ = 1.
On the other hand, by a direct computation we obtain

K(X, Y ) =
⟨
RRn1+pp

(X1, Y1)X1, Y1
⟩
Rn1+pp

+
⟨
RNn2κ (X2, Y2)X2, Y2

⟩
N

n2
κ

(3.6)

for every X = (X1, X2), Y = (Y1, Y2) ∈ TM such that ⟨X, Y ⟩ = 0, ⟨X, X⟩ =
⟨Y, Y ⟩ = 1.

Consequently, from (3.6) we get

K(X, Y ) = κ
(
|X2|2|Y2|2 − ⟨X2, Y2⟩2

)
≥ min{κ, 0}. (3.7)

Moreover, we have that

K(ξi, ξj) = 0, for all i, j ∈ {1, . . . , p} (3.8)

and ⟨
R(ξi, X)ξj, X

⟩
= 0, for all i, j ∈ {1, . . . , p}. (3.9)

We observe from (3.5), (3.7), (3.8) and (3.9) that the curvature constraints
(3.1), (3.2), (3.3) and (3.4) are satisfied with c1 = c3 = 0 and c2 ≤ min{κ, 0}.

Denote by RCD the components of the Ricci tensor of L
n+p
p , then the scalar

curvature R of L
n+p
p is given by

R =
∑
A

εARAA =
∑
i,j

Rijij − 2
∑
i,α

Riαiα +
∑
α,β

Rαβαβ.

If L
n+p
p satisfies conditions (3.1) and (3.4), then

R =
∑
i,j

Rijij − 2pc1 + (p − 1)c3. (3.10)

But, it is well known that the scalar curvature of a locally symmetric Lorentz
space is constant. Consequently,

∑
i,j Rijij is a constant naturally attached

to a locally symmetric Lorentz space satisfying conditions (3.1) and (3.4).
For sake of simplicity, in the course of this work we will denote the constant

1
n(n−1)

∑
i,j Rijij by R. In order to establish our main results, we devote this

section to present some auxiliary lemmas. Using the ideas of the Proposi-
tion 2.2 of [19] we have



62 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

Lemma 1. Let Mn be a linear Weingarten spacelike submanifold immersed
in locally symmetric space L

n+p
p satisfying conditions (3.1) and (3.4), such that

R = aH + b for some a, b ∈ R. Suppose that

(n − 1)a2 + 4n
(
R − b

)
≥ 0. (3.11)

Then,

|∇B|2 ≥ n2|∇H|2. (3.12)

Moreover, if the equality holds in (3.12) on Mn, then H is constant on Mn.

Proof. Since we are supposing that R = aH + b and L
n+p
p satisfies the

conditions (3.1) and (3.4) then from equation (2.6) we get

2
∑
i,j,α

hαijh
α
ijk =

(
2n2H + n(n − 1)a

)
Hk, (3.13)

where Hk stands for the k-th component of ∇H.
Thus,

4
∑
k

(∑
i,j,α

hαijh
α
ijk

)2
=
(
2n2H + n(n − 1)a

)2|∇H|2.
Consequently, using Cauchy-Schwarz inequality, we obtain that

4S|∇B|2 = 4
∑
i,j,α

(
hαij
)2 ∑

i,j,k,α

(
hαijk

)2 ≥ 4∑
k

(∑
i,j,α

hαijh
α
ijk

)2
=
(
2n2H + n(n − 1)a

)2|∇H|2.
(3.14)

On the other hand, since R = aH + b, from equation (2.6) we easily see that(
2n2H + n(n − 1)a

)2
= n2(n − 1)

[
(n − 1)a2 + 4n

(
R − b

)]
+ 4n2S. (3.15)

Thus, from (3.14) and (3.15) we have

4S|∇B|2 ≥ n2(n − 1)
[
(n − 1)a2 + 4n

(
R − b

)]
+ 4n2S|∇H|2, (3.16)

and taking account that since (n − 1)a2 + 4n
(
R − b

)
≥ 0, from (3.16) we

obtain

S|∇B|2 ≥ Sn2|∇H|2.



linear weingarten spacelike submanifolds 63

Therefore, either S = 0 and |∇B|2 = n2|∇H|2 = 0 or |∇B|2 ≥ n2|∇H|2.
Now suppose that |∇B|2 = n2|∇H|2. If (n − 1)a2 + 4n

(
R − b

)
> 0 then

from (3.16) we have that H is constant. If (n − 1)a2 + 4n
(
R − b

)
= 0, then

from (3.15) (
2n2H + n(n − 1)a

)2 − 4n2S = 0. (3.17)
This together with (3.13) forces that

S2k = 4n
2SH2k, k = 1, . . . , n, (3.18)

where Sk stands for the k-th component of ∇S.
Since the equality in (3.14) holds, there exists a real function ck on M

n

such that

hn+1ijk = ckh
n+1
ij ; h

α
ijk = ckh

α
ij, α > n + 1; i, j, k = 1, . . . , n. (3.19)

Taking the sum on both sides of equation (3.19) with respect to i = j, we get

Hk = ckH; H
α
k = 0, α > n + 1; k = 1, . . . , n. (3.20)

From second equation in (3.20) we can see that en+1 is parallel. It follows
from (3.19) that

Sk = 2
∑
i,j,k,α

hαijh
α
ijk = 2ckS, k = 1, . . . , n. (3.21)

Multiplying both sides of equation (3.21) by H and using (3.20) we have

HSk = 2HkS, k = 1, . . . , n. (3.22)

It follows from (3.18) and (3.22) that

H2kS = H
2
kn

2H2, k = 1, . . . , n. (3.23)

Hence we have

|∇H|2
(
S − n2H2

)
= 0. (3.24)

We suppose that H is not constant on Mn. In this case, |∇H| is not vanishing
identically on Mn. Denote M0 = {x ∈ M; |∇H| > 0} and T = S − n2H2.
It follows form (3.24) that M0 is open in M and T = 0 over M0. From
the continuity of T , we have that T = 0 on the closure cl(M0) of M0. If
M − cl(M0) ̸= ∅, then H is constant in M − cl(M0). It follows that S is
constant and hence T is constant in M − cl(M0). From the continuity of T ,
we have that T = 0 and hence S = n2H2 on Mn. It follows that H is constant
on Mn, which contradicts the assumption. Hence we complete the proof.



64 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

In our next result, we will deal with submanifolds Mn of L
n+p
p having

parallel normalized mean curvature vector field, which means that the mean
curvature function H is positive and that the corresponding normalized mean
curvature vector field H

H
is parallel as a section of the normal bundle. Extend-

ing the ideas of [9] we obtain the following Simons type formula for locally
symmetric spaces.

Lemma 2. Let Mn be an n-dimensional (n ≥ 2) submanifold with flat
normal bundle and parallel normalized mean curvature vector field in a locally
symmetric semi-Riemannian space L

n+p
p . Then, we have

1

2
∆S = |∇B|2 + 2

( ∑
i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α

hαijh
α
jmRmkik

)
+

∑
i,j,k,α,β

hαijh
β
jkRαiβk −

∑
i,j,k,α,β

hαijh
β
jkRαkβi

+
∑

i,j,k,α,β

hαijh
β
ijRαkβk −

∑
i,j,k,α,β

hαijh
β
kkRαiβj

+ n
∑
i,j

hn+1ij Hij − nH
∑

i,j,m,α

hαijh
α
mih

n+1
mj

+
∑
α,β

[
tr
(
hαhβ

)]2
+

3

2

∑
α,β

N
(
hαhβ − hβhα

)
,

(3.25)

where N(A) = tr(AAt), for all matrix A = (aij).

Proof. Note that

1

2
∆S =

∑
i,j,α

hαij∆h
α
ij +

∑
i,j,k,α

(
hαijk

)2
.

Using the definition of ∆hαij =
∑

k h
α
ijkk and the fact that |∇B|

2 =
∑

i,j,k(h
α
ijk)

2

we have
1

2
∆S =

∑
i,j,k,α

hαijh
α
ijkk + |∇B|

2.

Using the Codazzi equation (2.9) and the fact that hαij = h
α
ji we get

1

2
∆S =

∑
i,j,k,α

hαijRαijkk +
∑
i,j,k,α

hαijh
α
kijk + |∇B|

2.



linear weingarten spacelike submanifolds 65

From (2.10) we obtain

1

2
∆S = |∇B|2 +

∑
i,j,k,α

hαijRαijkk +
∑
i,j,k,α

hαijh
α
kikj +

∑
i,j,k,m,α

hαijh
α
kmRmijk+

+
∑

i,j,k,m,α

hαijh
α
miRmkjk.

Thence,

1

2
∆S = |∇B|2 +

∑
i,j,k,α

hαijRαijkk +
∑
i,j,k,α

hαijh
α
kkij +

∑
i,j,k,α

hαijRαkikj+

+
∑

i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α

hαijh
α
miRmkjk.

Using the Gauss equation (2.5) we get∑
i,j,k,m,α

hαijh
α
kmRmijk =

∑
i,j,k,m,α

hαijh
α
kmRmijk −

∑
i,j,k,m,α,β

hαijh
α
kmh

β
mjh

β
ik+

+
∑

i,j,k,m,α,β

hαijh
α
kmh

β
mkh

β
ij

and ∑
i,j,k,m,α

hαijh
α
miRmkjk =

∑
i,j,k,m,α

hαijh
α
miRmkjk − n

∑
i,j,m,α,β

hαijh
α
mih

β
mjH

β+

+
∑

i,j,k,m,α,β

hαijh
α
mih

β
mkh

β
kj.

Since we can choose a local orthonormal frame {e1, . . . , en+p} such that
en+1 =

H
H
, we have that

Hn+1 =
1

n
tr(hn+1) = H and Hα =

1

n
tr(hα) = 0, for α ≥ n + 2.

Thus, we get∑
i,j,k,m,α

hαijh
α
miRmkjk =

∑
i,j,k,m,α

hαijh
α
miRmkjk − n

∑
i,j,m,α

hαijh
α
mih

n+1
mj H+

+
∑

i,j,k,m,α,β

hαijh
α
mih

β
mkh

β
kj.



66 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

Therefore, ∑
i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α

hαijh
α
miRmkjk

=
∑

i,j,k,m,α

hαijh
α
kmRmijk −

∑
i,j,k,m,α,β

hαijh
α
kmh

β
mjh

β
ik

+
∑

i,j,k,m,α,β

hαijh
α
kmh

β
mkh

β
ij +

∑
i,j,k,m,α

hαijh
α
miRmkjk

− n
∑

i,j,m,α

hαijh
α
mih

n+1
mj H +

∑
i,j,k,m,α,β

hαijh
α
mih

β
mkh

β
kj.

From (2.11) we have∑
i,j,k,α

hαijRαijkk =
∑
i,j,k,α

hαijRαijk;k +
∑

i,j,k,α,β

hαijh
β
ikRαβjk +

∑
i,j,k,α,β

hαijh
β
jkRαiβk

+
∑

i,j,k,α,β

hαijh
β
kkRαijβ +

∑
i,j,k,m,α

hαijh
α
kmRmijk.

Using the Ricci equation (2.7) , we conclude that∑
i,j,k,α,β

hαijh
β
ikRαβjk =

∑
i,j,k,m,α,β

hαijh
β
ikh

α
jmh

β
mk −

∑
i,j,k,m,α,β

hαijh
β
ikh

α
mkh

β
jm.

Thence,∑
i,j,k,α

hαijRαijkk =
∑
i,j,k,α

hαijRαijk;k +
∑

i,j,k,m,α,β

hαijh
β
ikh

α
jmh

β
mk

−
∑

i,j,k,m,α,β

hαijh
β
ikh

α
mkh

β
jm +

∑
i,j,k,α,β

hαijh
β
jkRαiβk

+
∑

i,j,k,α,β

hαijh
β
kkRαijβ +

∑
i,j,k,m,α

hαijh
α
kmRmijk.

On other hand∑
i,j,k,α

hαijRαkikj =
∑
i,j,k,α

hαijRαkik;j +
∑

i,j,k,α,β

hαijh
β
kjRαβik +

∑
i,j,k,α,β

hαijh
β
ijRαkβk

+
∑

i,j,k,α,β

hαijh
β
kjRαkiβ +

∑
i,j,k,m,α

hαijh
α
jmRmkik.



linear weingarten spacelike submanifolds 67

Thence,

∑
i,j,k,α,β

hαijh
β
kjRαβik =

∑
i,j,k,m,α,β

hαijh
β
kjh

α
imh

β
mk −

∑
i,j,k,m,α,β

hαijh
β
kjh

α
mkh

β
im.

Therefore,

∑
i,j,k,α

hαijRαkikj =
∑
i,j,k,α

hαijRαkik;j +
∑

i,j,k,m,α,β

hαijh
β
kjh

α
imh

β
mk

−
∑

i,j,k,m,α,β

hαijh
β
kjh

α
mkh

β
im +

∑
i,j,k,α,β

hαijh
β
ijRαkβk

+
∑

i,j,k,α,β

hαijh
β
kjRαkiβ +

∑
i,j,k,m,α

hαijh
α
jmRmkik.

Hence,

∑
i,j,k,α

hαij(Rαijkk + Rαkikj) =
∑
i,j,k,α

hαij(Rαijk;k + Rαkik;j)

+
∑

i,j,k,m,α,β

hαijh
β
ikh

α
jmh

β
mk −

∑
i,j,k,m,α,β

hαijh
β
ikh

α
mkh

β
jm

+
∑

i,j,k,α,β

hαijh
β
jkRαiβk +

∑
i,j,k,α,β

hαijh
β
kkRαijβ

+
∑

i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α,β

hαijh
β
kjh

α
imh

β
mk

−
∑

i,j,k,m,α,β

hαijh
β
kjh

α
mkh

β
im +

∑
i,j,k,α,β

hαijh
β
ijRαkβk

+
∑

i,j,k,α,β

hαijh
β
kjRαkiβ +

∑
i,j,k,m,α

hαijh
α
jmRmkik.

Since L
n+p
p is locally symmetric, we have that

∑
i,j,k,α

hαij(Rαijk;k + Rαkik;j) = 0.



68 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

Thus,

∑
i,j,k,α

hαij(Rαijkk + Rαkikj) =
∑

i,j,k,m,α,β

hαijh
β
ikh

α
jmh

β
mk −

∑
i,j,k,m,α,β

hαijh
β
ikh

α
mkh

β
jm

+
∑

i,j,k,α,β

hαijh
β
jkRαiβk +

∑
i,j,k,α,β

hαijh
β
kkRαijβ

+
∑

i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α,β

hαijh
β
kjh

α
imh

β
mk

−
∑

i,j,k,m,α,β

hαijh
β
kjh

α
mkh

β
im +

∑
i,j,k,α,β

hαijh
β
ijRαkβk

+
∑

i,j,k,α,β

hαijh
β
kjRαkiβ +

∑
i,j,k,m,α

hαijh
α
jmRmkik.

Now, observe that ∑
i,j,k,α

hαijh
α
kkij = n

∑
i,j,α

hαijH
α
ij.

Using the fact that Hkj = H
n+1
kl and H

α
kj = 0, for α > n + 1 we have

∑
i,j,k,α

hαijh
α
kkij = n

∑
i,j

hn+1ij Hij.

Finally, we conclude that

1

2
∆S = |∇B|2 +

∑
i,j,k,m,α,β

hαijh
β
ikh

α
jmh

β
mk −

∑
i,j,k,m,α,β

hαijh
β
ikh

α
mkh

β
jm

+
∑

i,j,k,α,β

hαijh
β
jkRαiβk + nH

∑
i,j,k,α

hαijRαijn+1

+
∑

i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α,β

hαijh
β
kjh

α
imh

β
mk

−
∑

i,j,k,m,α,β

hαijh
β
kjh

α
mkh

β
im +

∑
i,j,k,α,β

hαijh
β
ijRαkβk +



linear weingarten spacelike submanifolds 69

+
∑

i,j,k,α,β

hαijh
β
kjRαkiβ +

∑
i,j,k,m,α

hαijh
α
jmRmkik + n

∑
i,j

hn+1ij Hij

+
∑

i,j,k,m,α

hαijh
α
kmRmijk −

∑
i,j,k,m,α,β

hαijh
α
kmh

β
mjh

β
ik

+
∑

i,j,k,m,α,β

hαijh
α
kmh

β
mkh

β
ij +

∑
i,j,k,m,α

hαijh
α
miRmkjk

− nH
∑

i,j,m,α

hαijh
α
mih

n+1
mj +

∑
i,j,k,m,α,β

hαijh
α
mih

β
mkh

β
kj.

(3.26)

Note that ∑
i,j,k,m,α,β

hαijh
β
ikh

α
jmh

β
mk −

∑
i,j,k,m,α,β

hαijh
β
ikh

α
mkh

β
jm

=
∑
α,β

tr
(
hαhβhβhα

)
−
∑
α,β

tr
(
hαhβ

)2
,

(3.27)

∑
i,j,k,m,α,β

hαijh
α
kmh

β
mkh

β
ij −

∑
i,j,k,m,α,β

hαijh
α
kmh

β
mjh

β
ik

=
∑
α,β

[
tr
(
hαhβ

)]2 − ∑
α,β

tr
(
hαhβ

)2
,

(3.28)

∑
i,j,k,m,α,β

hαijh
β
kjh

α
imh

β
mk =

∑
α,β

tr
(
hαhβhβhα

)
(3.29)

and ∑
i,j,k,m,α,β

hαijh
α
mih

β
mkh

β
kj −

∑
i,j,k,m,α,β

hαijh
β
kjh

α
mkh

β
im

=
1

2

∑
α,β

N
(
hαhβ − hβhα

)
.

(3.30)

Therefore, inserting (3.27), (3.28), (3.29) and (3.30) into (3.26) we com-
plete the proof.

In order to study linear Weingarten submanifolds, we will consider, for
each a ∈ R, an appropriated Cheng-Yau’s modified operator, which is given
by

L = � +
n − 1
2

a∆, (3.31)



70 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

where, according to [7], the square operator is defined by

�f =
∑
i,j

(
nHδij − nhn+1

)
fij, (3.32)

for each f ∈ C∞(M), and the normal vector field en+1 is taken in the direction
of the mean curvature vector field, that is, en+1 =

H
H
.

The next lemma guarantees us the existence of an Omori-type sequence
related to the operator L.

Lemma 3. Let Mn be a complete linear Weingarten spacelike in a locally
symmetric semi-Riemannian space L

n+p
p (c) satisfying conditions (3.1), (3.3)

and (3.4), such that R = aH + b, with a ≥ 0 and (n − 1)a2 + 4n
(
R − b

)
≥ 0.

If H is bounded on Mn, then there is a sequence of points {qk}k∈N ⊂ Mn
such that

lim
k

nH(qk) = sup
M

nH, lim
k

|∇nH(qk)| = 0 and lim sup
k

L(nH(qk)) ≤ 0.

Proof. Let us choose a local orthonormal frame {e1, . . . , en} on Mn such
that hn+1ij = λ

n+1
i δij. From (3.31) we have that

L(nH) = n
∑
i

(
nH +

n − 1
2

a − λn+1i

)
Hii.

Thus, for all i = 1, . . . , n and since that L
n+p
p satisfies the conditions (3.1) and

(3.4) then from (2.6) and with straightforward computation we get

(λn+1i )
2 ≤ S = n2H2 + n(n − 1)

(
aH + b − R

)
=

(
nH +

n − 1
2

a

)2
−

n − 1
4

[
(n − 1)a2 + 4n

(
R − b

)]
≤
(
nH +

n − 1
2

a

)2
,

where we have used our assumption that (n−1)a2 +4n
(
R − b

)
≥ 0 to obtain

the last inequality. Consequently, for all i = 1, . . . , n, we have

|λn+1i | ≤
∣∣∣∣nH + n − 12 a

∣∣∣∣ . (3.33)



linear weingarten spacelike submanifolds 71

Thus, from (2.6) we obtain

Rijij = Rijij −
∑
α

hαiih
α
jj +

∑
α

(hαij)
2

≥ Rijij −
∑
α

hαiih
α
jj.

Since

S ≤
(
nH +

n − 1
2

a

)2
,

we get that

(hαij)
2 ≤

(
nH +

n − 1
2

a

)2
,

for every α, i, j and, hence, from (3.33) we have

hαiih
α
jj ≤ |h

α
ii||h

α
jj| ≤

(
nH +

n − 1
2

a

)2
.

Therefore, since we are supposing that H is bounded on Mn and L
n+p
p satisfies

the condition (3.3), this is, Rijij ≥ c2, it follows that the sectional curvatures of
Mn are bounded from below. Thus, we may apply the well known generalized
maximum principle of Omori [17] to the function nH, obtaining a sequence
of points {qk}k∈N in Mn such that

lim
k

nH(qk) = sup nH, lim
k

|∇nH(qk)| = 0, and

lim sup
k

∑
i

nHii(qk) ≤ 0.
(3.34)

Since supM H > 0, taking subsequences if necessary, we can arrive to a se-
quence {qk)k∈N in Mn which satisfies (3.34) and such that H(qk) ≥ 0. Hence,
since a ≥ 0, we have

0 ≤ nH(qk) +
n − 1
2

a − |λn+1i (qk)| ≤ nH(qk) +
n − 1
2

a − λn+1i (qk)

≤ nH(qk) +
n − 1
2

a + |λn+1i (qk)| ≤ 2nH(qk) + (n − 1)a.



72 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

This previous estimate shows that function nH(qk) +
n−1
2

a − λn+1i (qk) is non-
negative and bounded on Mn, for all k ∈ N. Therefore, taking into account
(3.34), we obtain

lim sup
k

(L(nH)(qk)) ≤ n
∑
i

lim sup
k

[(
nH +

n − 1
2

a − λn+1i

)
(qk)Hii(qk)

]
≤ 0.

We close this section with the following algebraic lemma, whose proof can
be found in [18].

Lemma 4. Let A, B : Rn −→ Rn be symmetric linear maps such that
AB − BA = 0 and tr(A) = tr(B) = 0. Then

∣∣tr(A2B)∣∣ ≤ n − 2√
n(n − 1)

N(A)
√

N(B),

where N(A) = tr(AAt), for all matrix A = (aij). Moreover, the equality holds
if and only if (n − 1) of the eigenvalues xi of B and corresponding eigenvalues
yi of A satisfy

|xi| =

√
N(B)

n(n − 1)
, xiyi ≥ 0 and yi =

√
N(A)

n(n − 1)

(
resp. −

√
N(A)

n(n − 1)

)
.

4. Main results

As before, the normal vector field en+1 is taken in the direction of the mean
curvature vector field, that is, en+1 =

H
H
. In this setting, we will consider the

following symmetric tensor

Φ =
∑
i,j,α

Φαijωi ⊗ ωjeα,

where Φn+1ij = h
n+1
ij − Hδij and Φ

α
ij = h

α
ij, n + 2 ≤ α ≤ n + p. Let

|Φ|2 =
∑

i,j,α(Φ
α
ij)

2 be the square of the length of Φ.

Remark 1. Since the normalized mean curvature vector of Mn is parallel,
we have ωn+1α = 0, for α > n+1. Thus, from of the structure equations of the
normal bundle of Mn, it follows that Rn+1βij = 0, for all α, i, j. Hence, from
Ricci equation, we have that hn+1hα−hαhn+1 = 0, for all α. This implies that



linear weingarten spacelike submanifolds 73

the matrix hn+1 commutes with all the matrix hα. Thus, being Φα = (Φαij),

we have that Φα = hα − Hα and, hence Φn+1 = hn+1 − Hn+1 and Φα = hα,
for α > n + 1. These form, Φn+1 commutes with all the matrix Φα. Since the
matrix Φα is traceless and symmetric, once the matrix hα are symmetric, we
can use Lemma 4 for the matrix Φα and Φn+1 in order to obtain

∣∣tr((Φα)2Φn+1)∣∣ ≤ n − 2√
n(n − 1)

N(Φα)
√

N(Φn+1). (4.1)

Summing (4.1) in α, we have

∑
α

∣∣tr((Φα)2Φn+1)∣∣ ≤ n − 2√
n(n − 1)

∑
α

N(Φα)
√
N(Φn+1).

In order to prove our characterization results, it will be essential the fol-
lowing lower boundedness for the Laplacian operator acting on the square of
the length of the second fundamental form. If L

n+p
p is a space form then from

[8] follows that R⊥ = 0 if and only if there exists an orthogonal basis for TM
that diagonalizes simultaneously all Bξ, ξ ∈ TM⊥.

Proposition 1. Let Mn be a linear Weingarten spacelike submanifold in
a semi-Riemannian locally symmetric space L

n+p
p satisfying conditions (3.1),

(3.2), (3.3) and (3.4), with parallel normalized mean curvature vector field
and flat normal bundle. Suppose that there exists an orthogonal basis for
TM that diagonalizes simultaneously all Bξ, ξ ∈ TM⊥. If Mn is such that
R = aH + b, with (n − 1)a2 + 4n

(
R − b

)
≥ 0 and c = c1

n
+ 2c2, then

L(nH) ≥ |Φ|2
(
|Φ|2

p
−

n(n − 2)√
n(n − 1)

H|Φ| − n(H2 − c)

)
.

Proof. Let us consider {e1, . . . , en} a local orthonormal frame on Mn such
that hαij = λ

α
i δij, for all α ∈ {n + 1, . . . , n + p}. From (3.25), we get

2

( ∑
i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α

hαijh
α
jmRmkik

)

= 2
∑
i,k,α

(
(λαi )

2Rikik + λ
α
i λ

α
k Rkiik

)
=
∑
i,k,α

Rikik(λ
α
i − λ

α
k )

2.



74 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

Since that L
n+p
p satisfies the condition (3.3) we have

2

( ∑
i,j,k,m,α

hαijh
α
kmRmijk +

∑
i,j,k,m,α

hαijh
α
jmRmkik

)

≥ c2
∑
i,k,α

(
λαi − λ

α
k

)2
= 2nc2|Φ|2.

(4.2)

Now, for each α, consider hα the symmetric matrix (hαij), and

Sαβ =
∑
i,j

hαijh
β
ij.

Then the (p × p)-matrix (Sαβ) is symmetric and we can see that is diagonal-
izable for a choose of en+1, . . . , en+p. Thence,

Sα = Sαα =
∑
i,j

hαijh
α
ij,

and we have that
S =

∑
α

Sα.

Since that L
n+p
p satisfies the condition (3.2) we obtain∑

i,j,k,α,β

hαijh
β
jkRαiβk −

∑
i,j,k,α,β

hαijh
β
jkRαkβi +

∑
i,j,k,α,β

hαijh
β
ijRαkβk

−
∑

i,j,k,α,β

hαijh
β
kkRαiβj =

∑
i,k,α

(λαi )
2Rαkαk − nH2c1.

Since that L
n+p
p satisfies the condition (3.1) we conclude that∑

i,j,k,α,β

hαijh
β
jkRαiβk −

∑
i,j,k,α,β

hαijh
β
jkRαkβi +

∑
i,j,k,α,β

hαijh
β
ijRαkβk

−
∑

i,j,k,α,β

hαijh
β
kkRαiβj = c1|Φ|

2.
(4.3)

Finally note that ∑
α,β

N
(
hαhβ − hβhα

)
≥ 0. (4.4)



linear weingarten spacelike submanifolds 75

Therefore, from (3.25) and using (4.2), (4.3) and (4.4) we conclude that

1

2
∆S ≥ |∇B|2 + cn|Φ|2 + n

∑
i,j

hn+1ij Hij

− nH
∑

i,j,m,α

hαijh
α
mih

n+1
mj +

∑
α,β

[
tr
(
hαhβ

)]2
.

(4.5)

From (3.31) we have

L(nH) = �(nH) +
n − 1
2

a∆(nH)

=
∑
i,j

(
nHδij − hn+1ij

)
(nH)ij +

n − 1
2

a∆(nH)

= n2H
∑
i

Hii − n
∑
i,j

hn+1ij Hij +
n − 1
2

a∆(nH)

= n2H∆H − n
∑
i,j

hn+1ij Hij +
n − 1
2

a∆(nH).

Note that
∆H2 = 2H∆H + 2|∇H|2.

Thus,

L(nH) =
1

2
∆
(
n2H2

)
− n2|∇H|2 − n

∑
i,j

hn+1ij Hij +
n − 1
2

a∆(nH).

Since that R = aH + b and L
n+p
p satisfies the conditions (3.1) and (3.3) we

have that R is constant then from (2.6) we get
1

2
n(n − 1)∆(aH) +

1

2
∆
(
n2H2

)
=

1

2
∆S.

Therefore, using the inequality (4.5) and Lemma 1 we conclude that

L(nH) =
1

2
∆S − n2|∇H|2 − n

∑
i,j

hn+1ij Hij

≥ |∇B|2 − n2|∇H|2 + cn|Φ|2

− nH
∑

i,j,m,α

hαijh
α
mih

n+1
mj +

∑
α,β

[
tr
(
hαhβ

)]2
≥ cn|Φ|2 − nH

∑
i,j,m,α

hαijh
α
mih

n+1
mj +

∑
α,β

[
tr
(
hαhβ

)]2
.



76 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

On the other hand, with a straightforward computation we guarantee that

− nH
∑
α

tr
[
hn+1

(
hα
)2]

+
∑
α,β

[
tr
(
hαhβ

)]2
= −nH

∑
α

tr
[
Φn+1

(
Φα
)2] − nH2|Φ|2 + ∑

α,β

[
tr
(
ΦαΦβ

)]2
≥

−n(n − 2)√
n(n − 1)

H|Φ|3 − nH2|Φ|2 +
|Φ|4

p
.

(4.6)

Therefore,

L(nH) ≥ cn|Φ|2 −
n(n − 2)√
n(n − 1)

H|Φ|3 − nH2|Φ|2 +
|Φ|4

p

= |Φ|2PH,p,c(|Φ|),

(4.7)

where

PH,p,c(x) =
x2

p
−

n(n − 2)√
n(n − 1)

Hx − n
(
H2 − c

)
.

When c > 0, if H2 ≥ 4(n−1)c
Q(p)

, where Q(p) = (n − 2)2p + 4(n − 1), then the
polynomial PH,p,c defined by

PH,p,c(x) =
x2

p
−

n(n − 2)√
n(n − 1)

Hx − n(H2 − c)

has (at least) a positive real root given by

C(n, p, H) =

√
n

2
√
n − 1

(
p(n − 2)H +

√
pQ(p)H2 − 4p(n − 1)c

)
.

On the other hand, in the case that c ≤ 0, the same occurs without any
restriction on the values of the mean curvature function H. Now, we are in
position to present our first theorem.

Theorem 1. Let Mn be a complete linear Weingarten spacelike subman-
ifold in locally symmetric L

n+p
p satisfying conditions (3.1), (3.2), (3.3) and

(3.4), with parallel normalized mean curvature vector and flat normal bundle,
such that R = aH + b with a ≥ 0 and (n − 1)a2 + 4n

(
R − b

)
≥ 0. Suppose

that there exists an orthogonal basis for TM that diagonalizes simultaneously



linear weingarten spacelike submanifolds 77

all Bξ, ξ ∈ TM⊥. When c > 0, assume in addition that H2 ≥
4(n−1)c
Q(p)

. If H is

bounded on Mn and |Φ| ≥ C(n, p, sup H), then p = 1 and Mn is an isopara-
metric hypersurface with two distinct principal curvatures one of which is
simple.

Proof. Since we are assuming that a ≥ 0 and that inequality (3.11) holds,
we can apply Lemma 3 to the function nH in order to obtain a sequence of
points {qk}k∈N ⊂ Mn such that

lim
k

nH(qk) = sup
M

nH, and lim sup
k

L(nH)(qk) ≤ 0. (4.8)

Thus, from (4.7) and (4.8) we have

0 ≥ lim sup
k

L(nH)(qk) ≥ sup
M

|Φ|2Psup H,p,c
(
sup
M

|Φ|
)
. (4.9)

On the other hand, our hypothesis imposed on |Φ| guarantees us that
supM |Φ| > 0. Therefore, from (4.9) we conclude that

Psup H,p,c

(
sup
M

|Φ|
)

≤ 0. (4.10)

Suppose, initially, the case c > 0. From our restrictions on H and |Φ|, we have
that PH,p,c(|Φ|) ≥ 0, with PH,p,c(|Φ|) = 0 if, and only if, |Φ| = C(n, p, H).
Consequently, from (4.10) we get

sup
M

|Φ| = C(n, p, sup H).

Taking into account once more our restriction on |Φ|, we have that |Φ| is
constant on Mn. Thus, since Mn is a linear Weingarten submanifold, from
(3.11) we have that H is also constant on Mn. Hence, from (4.7) we obtain

0 = L(nH) ≥ |Φ|2PH,p,c(|Φ|) ≥ 0.

Since |Φ| > 0, we must have PH,p,c(|Φ|) = 0. Thus, all inequalities obtained
along the proof of Proposition 1 are, in fact, equalities. In particular, from
inequality (4.6) we conclude that

tr(Φn+1) = |Φ|2.

So, from (2.6) we get

tr(Φn+1)2 = |Φ|2 = S − nH2. (4.11)



78 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

On the other hand, we also have that

tr
(
Φn+1

)2
= S −

∑
α>n+1

∑
i,j

(
hαij
)2 − nH2. (4.12)

Thus, from (4.11) and (4.12) we conclude that
∑

α>n+1

∑
i,j(h

α
ij)

2 = 0. But,
from inequality (4.6) we also have that

|Φ|4 = p
∑
α

[
N
(
Φα
)]2

= pN
(
Φn+1

)2
= p|Φ|4. (4.13)

Hence, since |Φ| > 0, we must have that p = 1. In this setting, from (3.12)
and (4.13) we get∑

i,j,k

(
hn+1ijk

)2
= |∇B|2 = n2|∇H|2 = 0,

that is, hn+1ijk = 0 for all i, j. Hence, we obtain that M
n is an isoparametric

hypersurface of L
n+p
p .

When c ≤ 0, we proceed as before until reach equation (4.10) and, from
|Φ| ≥ C(n, p, sup H), we have that PH,p,c(|Φ|) ≥ 0. At this point, we can
reason as in the previous case to obtain that H is constant, p = 1 and,
consequently, we also conclude that Mn is an isoparametric hypersurface of
L
n+p
p . Hence, since the equality occurs in (4.1), we have that also occurs the

equality in Lemma 4. Consequently, Mn has at most two distinct constant
principal curvatures.

In particular, when the immersed submanifold has constant scalar curva-
ture, from Theorem 1 we obtain the following

Corollary 1. Let Mn be a complete spacelike submanifold in locally
symmetric semi-Riemannian L

n+p
p satisfying conditions (3.1), (3.2), (3.3) and

(3.4), with parallel normalized mean curvature vector, flat normal bundle
and constant normalized scalar curvature R satisfying R ≤ c. Suppose that
there exists an orthogonal basis for TM that diagonalizes simultaneously all
Bξ, ξ ∈ TM⊥. When c > 0, assume in addition that H2 ≥

4(n−1)c
Q(p)

. If

H is bounded on Mn and |Φ| ≥ C(n, p, sup H), then p = 1 and Mn is an
isoparametric hypersurface with two distinct principal curvatures one of which
is simple.



linear weingarten spacelike submanifolds 79

In order to establish our next theorem, we will need of the following lemma
obtained by Caminha, which can be regarded as an extension of Hopf’s maxi-
mum principle for complete Riemannian manifolds (cf. Proposition 2.1 of [5]).
In what follows, let L1(M) denote the space of Lebesgue integrable functions
on Mn.

Lemma 5. Let X be a smooth vector field on the n-dimensional complete
noncompact oriented Riemannian manifold Mn, such that divMX does not
change sign on Mn. If |X| ∈ L1(M), then divMX = 0.

We close our paper stating and proving our second characterization theo-
rem.

Theorem 2. Let Mn be a complete linear Weingarten spacelike submani-
fold in locally symmetric Einstein semi-Riemannian L

n+p
p satisfying conditions

(3.1), (3.2), (3.3) and (3.4), with parallel normalized mean curvature vector,
flat normal bundle such that R = aH +b, with (n−1)a2 +4n(R−b) ≥ 0. Sup-
pose that there exists an orthogonal basis for TM that diagonalizes simulta-
neously all Bξ, ξ ∈ TM⊥. When c > 0, assume in addition that H2 ≥

4(n−1)c
Q(p)

.

If H is bounded on Mn, |Φ| ≥ C(n, p, H) and |∇H| ∈ L1(M), then p = 1 and
Mn is a isoparametric hypersurface with two distinct principal curvatures one
of which is simple.

Proof. Since the ambient space L
n+p
p is supposed to be Einstein, reasoning

as in the first part of the proof of Theorem 1.1 in [10], from (3.31) and (3.32)
it is not difficult to verify that

L(nH) = divM(P(∇H)), (4.14)

where

P =

(
n2H +

n(n − 1)
2

a

)
I − nhn+1. (4.15)

On the other hand, since R = aH + b and H is bounded on Mn, from equa-
tion (2.6) we have that B is bounded on Mn. Consequently, from (4.15) we
conclude that the operator P is bounded, that is, there exists C1 such that
|P | ≤ C1. Since we are also assuming that |∇H| ∈ L1(M), we obtain that

|P(∇H)| ≤ |P ||∇H| ≤ C1|∇H| ∈ L1(M). (4.16)



80 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez

So, from Lemma 5 and (4.14) we obtain that L(nH) = 0 on Mn. Thus,

0 = L(nH) ≥ |Φ|2PH,p,c(|Φ|) ≥ 0 (4.17)

and, consequently, we have that all inequalities are, in fact, equalities. In
particular, from (3.11) we obtain

|∇B|2 = n2|∇H|2. (4.18)

Hence, Lemma 1 guarantees that H is constant. At this point, we can proceed
as in the last part of the proof of Theorem 1 to conclude our result.

Acknowledgements

The first author is partially supported by CAPES, Brazil. The sec-
ond author is partially supported by CNPq, Brazil, grant 303977/2015-
9. The fourth author is partially supported by CNPq, Brazil, grant
308757/2015-7. The authors would like to thank the referee for reading
the manuscript in great detail and for his/her valuable suggestions and
useful comments.

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