BIHARMONIC PSEUDO-RIEMANNIAN SUBMERSIONS FROM
3-MANIFOLDS
İREM KÜPELI ERKEN AND CENGIZHAN MURATHAN
Abstract. We classify the pseudo-Riemannian biharmonic submersion from a 3-
dimensional space form into a surface.
1. INTRODUCTION
The theory of Riemannian submersions was initiated by O’Neill [14] and Gray [11].
One of the well known example of a Riemannian submersion is the projection of a Rie-
mannian product manifold on one of its factors. Presently, there is an extensive lit-
erature on the Riemannian submersions with different conditions imposed on the total
space and on the fibres. A systematic exposition could be found in A. Besse’s book
[4]. Pseudo-Riemannian submersions were introduced by O’Neill [15]. Magid classified
pseudo-Riemannian submersions with totally geodesic fibres from an anti-de Sitter space
onto a Riemannian manifold [13]. Then Bădiţou gave the classification of the pseudo-
Riemannian submersions with (para) complex connected totally geodesic fibres from a
(para) complex pseudo-hyperbolic space onto a pseudo Riemannian manifold [1, 3].
A map between Riemannian manifolds is harmonic if the divergence of its differential
vanishes. The first major study of harmonic maps has been begun by J. Eells and J. H.
Sampson [9]. In [9], Eells and Sampson defined biharmonic maps between Riemannian
manifolds as an extension of harmonic maps and Jiang obtained their first and second
variational formulas [12].
During the last decade important progress has been made in the study of both the
geometry and the analytic properties of biharmonic maps. A fundamental problem in
the study of biharmonic maps is to classify all proper biharmonic maps between certain
model spaces. An example of this was proved independently by Chen-Ishikawa [7] and
Jiang [12] that every biharmonic surface in a Euclidean 3-space E3 is a minimal surface.
Later, Caddeo et al. showed that the theorem remains true if the target Euclidean space
is replaced by 3-dimensional hyperbolic space form [5]. Chen and Ishikawa also proved
that biharmonic Riemannian surface in E31 is a harmonic surface [6]. For Riemannian
submersions, Wang and Ou stated that Riemannian submersion from a 3-dimensional
space form into a surface is biharmonic if and only if it is harmonic [19].
The above results give us the motivation for preparing this study. In this paper, we
study the biharmonic pseudo-Riemannian submersions from 3-manifolds.
The main purpose of section §2 is to give a brief information about pseudo-Riemannian
submersions, biharmonic maps and space forms. In this section, we also give some proper-
ties of fundamental tensors and fundamental equations which we will use them to obtain
Date: 23.01.2017.
2000 Mathematics Subject Classification. Primary 53B20, 53B25, 53B50; Secondary 53C15, 53C25.
Key words and phrases. pseudo-Riemannian submersions, biharmonic 3-manifolds.
1
arXiv:1206.1768v3  [math.DG]  11 Feb 2017
2 İREM KÜPELI ERKEN AND CENGIZHAN MURATHAN
our results. In section §3, we investigate the biharmonicity of a pseudo-Riemannian sub-
mersion from a 3-manifold by using the integrability data of a special orthonormal frame
adapted to a pseudo-Riemannian submersion. Finally, we give a complete classification
of biharmonic pseudo-Riemannian submersions from a 3-dimensional pseudo-Riemannian
space form.
2. PRELIMINARIES
2.1. Pseudo-Riemannian submersions with totally geodesic fibre. In this sub-
section we recall several notions and results which will be needed throughout the paper.
Let (M, g) be an m-dimensional connected pseudo-Riemannian manifold of index s
(0 ≤ s ≤ m), let (B, g′) be an n-dimensional connected pseudo-Riemannian manifold
of index r ≤ s, (0 ≤ r ≤ n). In case of Riemannian submersion, the fibers are always
Riemannian manifolds.
A pseudo-Riemannian submersion is a smooth map π : M → B which is onto and
satisfies the following three axioms:
S1. π∗ |p is onto for all p ∈ M ,
S2. the restriction of the metric to the fibres π−1(b), b ∈ B are non degenerate ,
S3. π∗ preserves scalar products of vectors normal to fibres.
We shall always assume that the dimension of the fibres dimM - dimB is positive and
the fibres are connected. By S2, one can observe fibres as spacelike and timelike cases.
The vectors tangent to fibres are called vertical and those normal to fibres are called
horizontal. We denote by V the vertical distribution and byH the horizontal distribution.
The fundamental tensors of a submersion were defined by O’Neill ([14], [15]). They are
(1, 2)-tensors on M , given by the formulas:
(2.1) T (E,F ) = TEF = h∇νEνF + ν∇νEhF,
A(E,F ) = AEF = ν∇hEhF + h∇hEυF,
for any E, F ∈ X(M). Here ∇ denotes the Levi-Civita connection of (M, g). These
tensors are called integrability tensors for the pseudo-Riemannian submersions. We use
the h and ν letters to denote the orthogonal projections on the vertical and horizontal
distributions respectively. A vector field X on M is said to be basic if it is the unique
horizontal lift of a vector field X∗ on B, so that π∗(X) = X∗ is horizontal and π-related
to a vector field X∗ on B. It is easy to see that every vector field X∗ on B has a unique
horizontal lift X to M and X is basic. The following lemmas are well known (see [14],
[15]).
Lemma 1. Let π : (M, g) → (B, g′) be a pseudo-Riemannian submersion. If X, Y are
basic vector fields on M , then
i) g(X,Y ) = g′(X∗, Y∗) ◦ π,
ii) h[X,Y ] is basic and π-related to [X∗, Y∗],
B
iii) h(∇XY ) is a basic vector field corresponding to∇X Y∗ where∇B is the connection∗
on B.
iv) for any vertical vector field V , [X,V ] is vertical.
Lemma 2. For any U,W vertical and X,Y horizontal vector fields, the tensor fields T
and A satisfy
i)TUW = TWU ,
ii)A 1XY = −AY X = 2ν [X,Y ] .
BIHARMONIC PSEUDO-RIEMANNIAN SUBMERSIONS FROM 3-MANIFOLDS 3
Moreover, if X is basic and U is vertical then h(∇UX) = h(∇XU) = AXU. Notice that
T acts on the fibres as the second fundamental form of the submersion and restricted to
vertical vector fields and it can be easily seen that T = 0 is equivalent to the condition
that the fibres are totally geodesic.
We define the curvature tensor R of M by R(E,F ) = ∇E∇F − ∇F∇E − ∇[E,F ] for
any vector fields E, F on M . The pseudo-Riemannian curvature (0, 4)-tensor is defined
by
R(E,F,G,H) = g(R(E,F )G,H).
Let us recall the sectional curvature of pseudo-Riemannian manifolds for nondegener-
ate planes. Let M be a pseudo-Riemannian manifold and P be a non-degenerate tangent
plane to M at p. The number
g(R(X,Y )Y,X)
KX∧Y =
g(X,X)g(Y, Y )− g(X,Y )2
is independent on the choice of basis X,Y for P and is called the sectional curvature.
We use notation Rijkl = g(R(ei, ej)ek, el). Next, we can give the following lemma:
Lemma 3 ([15]). Let π : (M, g) → (B, g′) be a pseudo-Riemannian submersion. K and
KB denote the sectional curvatures in M and B, respectively. If X, Y are basic vector
fields on M, then
3g(AXY,AXY )
(2.2) KBX ∧Y = KX∧Y + .∗ ∗ g(X,X)g(Y, Y )− g(X,Y )2
In [17], Escobales gave a classification of Riemannian submersions with connected
totally geodesic fibres from a sphere to a Riemannian manifold and then Ranjan [16]
dropped Escobales’s classification into three categories: (a) S2n+1 → CPn, n ≥ 1, with
the fibres S1; (b) S4n+3 → HPn,n ≥ 1, with the fibres S3; (c) S8n+7 → CaPn, n = 1, 2
with the fibres S7, where CPn, HPn and CaPn are complex projective, quaternionic
projective and Cayley projective space, respectively.
In the Lorentz space case, Magid [13] proved that if π : H2n+1(c) → B2n1 be a
pseudo-Riemannian submersion with totally geodesic fibres onto a Riemannian mani-
fold then, B2n is a Kaehler manifold holomorphically isometric to complex hyperpolic
space CHn(4c).
In [2] Baditou and Ianuş generalizedMagid’s result and classified the pseudo-Riemannian
submersions with connected complex totally geodesic fibres from a complex pseudo hy-
perbolic space onto a Riemannian manifold. These pseudo-Riemannian submersions are
observed as mainly three categories : (1) H2m+1 m1 → CH , (2) H4m+33 → H(Hm) or
(3) H15 → H87 (−4), where mCH and H(Hm) are complex hyperbolic space and quater-
nionic hyperbolic space, respectively. Then Baditoiu [1] improved these results under the
assumption that the dimension of the fibres is less than or equal to three.
Recently, Baditoiu [3] generalized previous results without any assumption for dimen-
sion of the fibres and proved that any pseudo-Riemannian submersions with connected,
totally geodesic fibres from a real pseudo hyperbolic space onto a pseudo-Riemannian
manifold is equivalent to one of the (para) Hopf pseudo-Riemannian submersions: (i)
H2m+1 m2t+1 → CHt , 0 ≤ t ≤ m, (ii) H2m+1m → APm, (iii) H4m+3 m4t+3 → H(Ht ), 0 ≤ t ≤ m,
(iv) H4m+3 → BPm, (v) H15 → H8(−4), (vi) H15 → H82m+1 15 8 7 4 (−4) or (vii) H157 → H84 (−4),
where CHm mt and H(Ht ) are the indefinite complex and quaternionic pseudo-hyperbolic
spaces of holomorphic, respectively, quaternionic curvature −4; APm is the para-complex
projective space of real dimension 2m, signature (m,m) and para-holomorphic curvature
4 İREM KÜPELI ERKEN AND CENGIZHAN MURATHAN
−4; BPm is the para-quaternionic projective space of real dimension 4m, signature
(2m, 2m) and para-quaternionic curvature −4.
In summary, for three dimensional, these (para) pseudo-Riemannian submersions with
connected, totally geodesic fibres fall into one of the following cases:
(a1) π : S
3(1) → CP 1, (a 3 2 1 32) π : H1 (−1) → H (−4) = CH , (a3) π : H1 (−1) →
H21 (−4) = AH1, (a4) π : H3 2 13 (−1) → H2 (−4) = CH1
We will finish this subsection by the following Theorem of Uniqueness:
Theorem 1 ([3]). Let π , π : Ha1 2 l → B be two pseudo-Riemannian submersions with con-
nected, totally geodesic fibres from a pseudo-hyperbolic space onto a pseudo-Riemannian
manifold. Then there exists an isometry f : Hal → Hal such that π2 ◦ f = π1. In
particular, π1 and π2 are equivalent.
2.2. Biharmonic maps. Let Mm and Bn be pseudo-Riemannian manifolds of dimen-
sions m and n, respectively, and ϕ : Mm → Bn a smooth map. We denote by ∇M and
∇B the Levi-Civita connections on Mm and Bn, respectively. Then the tension field
τ(ϕ) is a section of the vector bundle ϕ∗TBn defined by
m
∑
τ (ϕ) = trace(∇ϕdϕ) = g(e , e )(∇ϕi i e dϕ(ei)− dϕ(∇e ei)).i i
i=1
Here ∇ϕ and {ei} denote the induced connection by ϕ on the bundle ϕ∗TBn, which is
the pull-back of ∇B, and a local orthonormal frame field of Mm, respectively. A smooth
map ϕ is called a harmonic map if its tension field vanishes. A map ϕ is called biharmonic
if it is a critical point of the energy
∫
1
E2(ϕ) = g(τ (ϕ), τ (ϕ)dvg
2 Ω
for every compact domains Ω of Mm, where dvg is the volume form of M
m. Using same
argument in Riemannian case, the bitension field can be defined by
m
∑
(2.3) τ ϕ ϕ ϕ B2(ϕ) = g(ei, ei)((∇e ∇e −∇ e )τ (ϕ)−R (dϕ(ei), τ (ϕ))dϕ(e∇ i)),i i ei i
i=1
where RB is the curvature tensor of Bn (see [8], [12], [18]). A smooth map ϕ is a
biharmonic map (or 2-harmonic map) if its bitension field vanishes (see [12], [18]). By
definition, a harmonic map is clearly biharmonic map. Non harmonic maps are called
proper biharmonic maps.
3. THE THEOREMS AND PROOFS
In this section, we will prove our classification Theorem and corollaries. Firstly, we
will recall well known theorems:
′
Theorem 2 ([10]). A pseudo-Riemannian submersion π : (M, g) → (B, g ) is a harmonic
map if and only if each fibre is a minimal submanifold.
′
Theorem 3 ([1],[13],[16],[17]). Let π : (M3(c), g) → (B2r s , g ) be a (para) pseudo-Riemannian
submersion with connected totally geodesic fibres, where 0 ≤ r ≤ 3, 0 ≤ s ≤ 2 and
c 6= 0.In summary, for three dimensional, these (para) pseudo-Riemannian submersions
with connected, totally geodesic fibres. Then π is one of the following types:
BIHARMONIC PSEUDO-RIEMANNIAN SUBMERSIONS FROM 3-MANIFOLDS 5
Timelike Fiber Spacelike Fiber
π π
H33 (−1) → H22 (−4) = CH11 ;[1] H31 (−1) → H21 (−4) = AH1;[1]
( )
H31 (−1) →
π
H2(−4) = CH1;[13] S3(1) →π S2 12 = CP 1;[16],[17].
We will report following theorems which give us the motivation to study on this paper.
Theorem 4 ([6]). Let x : M → E3s (s = 0, 1) be a biharmonic isometric immersion of
a Riemannian surface M into E3s .Then x is harmonic.
Theorem 5 ([20]). If M is a complete biharmonic space-like surface in S3 or R31 1, then
it must be totally geodesic, i.e. S2 or R2.
′
Theorem 6 ([19]). Let π : (M3(c), g) → (B2, g ) be a Riemannian submersion from a
space form of constant sectional curvature c. Then, π is biharmonic if and only if it is
harmonic, and this holds if and only if it is a harmonic morphism.
′
Let π : (M3r , g) → (B2s , g ) be a pseudo-Riemannian submersion where 0 ≤ r ≤ 3,
0 ≤ s ≤ 2. Let us consider a local pseudo orthonormal frame {e1, e2, e3} such that e1, e2
are basic and e3 is vertical . Then, it is well known (see [14]) that [e1, e3] and [e2, e3] are
vertical and [e1, e2] is π-related to [ε1, ε2], where {ε1, ε2} is a pseudo orthonormal frame
in the base manifold.
Let {e1, e2, e3} be an orthonormal frame adapted to with e3 being vertical where
g(ei, ei) = δi = ∓1. If we assume that
(3.1) [ε1, ε2] = L1ε1 + L2ε2,
for L , L ∈ C∞1 2 (B) and use the notations li = Li ◦ π, i = 1, 2. Then, we have
[e1, e3] = λe3,
(3.2) [e2, e3] = µe3,
[e1, e2] = l1e1 + l2e2 − 2σe3.
where λ, µ and σ ∈ C∞(M). Here l1, l2, λ, µ and σ are the integrability functions of the
adapted frame of the pseudo-Riemannian submersion π.
′
Proposition 1. Let π : (M3r , g) → (B2s , g ) be a pseudo-Riemannian submersion with
the adapted frame {e1, e2, e3} and the integrability functions l1, l2, λ, µ and σ. Then,
the pseudo-Riemannian submersion π is biharmonic if and only if
∆Mλ = −δ2l1e1(µ)− δ2e1(µl1)− δ2l2e2(µ)− δ2e2(µl2)
{ }
(3.3) +δ 2 2 2 B2λµl1 + δ2µ l2 + λ δ2l1 + δ1l2 − δ1δ2K ,
∆Mµ = δ1l1e1(λ) + δ1e1(λl1) + δ1l2e2(λ) + δ1e2(λl2)
{ }
−δ1λµl2 − δ 21λ l1 + µ δ l22 1 + δ l21 2 − δ δ B1 2K ,
where KB = RB1221 ◦ π = δ2e1(l2) − δ1e2(l1) − δ1l21 − δ2l22 is the Gauss curvature of
′
Riemannian manifold (B2s , g ).
Proof. Let ∇ denote the Levi-Civita connection of the pseudo-Riemannian manifold
(M3r , g). Using (3.2), Koszul formula and after a straightforward computation, we have
∇e e1 = −δ1δ2l1e2, ∇e e2 = l1e1 − σe3,1 1
∇e e3 = δ2δ3σe2, ∇1 e e2 1 = −l2e2 + σe3,
(3.4) ∇e e2 = δ1δ2l2e1, ∇e e3 = −δ1δ3σe ,2 2 1
∇e e1 = δ2δ3σe2 − λe3, ∇e e2 = −δ1δ3σe1 − µe3,3 3
∇e e3 = δ1δ3λe1 + δ2δ3µe3 2.
6 İREM KÜPELI ERKEN AND CENGIZHAN MURATHAN
The tension of the pseudo-Riemannian submersion τ is given by
3
∑
[ ]
(3.5) τ (π) = g(e , e ) ∇πi i e dπ(ei)− dπ(∇M Mi e ei) = −δ3dπ(∇e e3) = −δ1λε1 − δ2µε2.i 3
i=1
After some calculation by using (3.4) we get
3
{ }
∑
τ2(π) = g(e , e ) ∇π ∇π τ (π)−∇π Bi i e M τ (π)−R (dπ(ei), τ (π))dπ(ei)i ei ∇ ee ii
i=1
[ ]
∇πe (−δ1e1(λ)ε1 − δ λ∇π1 e ε1) +∇πe (−δ2e1(µ)ε π= 2 − δ2µ∇δ 1 1 1 e ε2)11 +δ δ l ∇π1 2 1 e (−δ1λε1 − δ2µε2) + δ µRB2 (ε1, ε2)ε12
[ ]
∇π (−δ e (λ)ε − δ λ∇π ε ) +∇πe 1 2 1 1 e 1 e (−δ2e2(µ)ε2 − δ2µ∇πe ε2)+δ 2 2 2 22 −δ1δ2l π2∇e (−δ1λε1 − δ2µε2) + δ1λRB(ε2, ε1)ε21
[ ]
∇πe (−δ1e3(λ)ε1 − δ λ∇π ε ) +∇π1 e 1 e (−δ2e π3(µ)ε2 − δ2µ∇e ε2)δ 3 3 3 33 −δ1δ3λ∇πe (−δ1λε1 − δ2µε2)− δ2δ µ∇π3 e (−δ1λε1 −
.
δ2µε2)
1 2
Now we calculate Laplace of λ and µ. Since gradλ = δ1e1(λ)e1 + δ2e2(λ)e2 + δ3e3(λ)e3,
we obtain
3
∑
∆mλ = g(ei, ei)g(∇e gradλ, ei i)
i=1
= δ1e1(e1(λ)) + δ2e2(e2(λ)) + δ3e3(e3(λ)) + δ2e2(λ)l1 − δ1e1(λ)l2
−δ1e1(λ)λ− δ2e2(λ)µ.
Using same calculations for µ we get
∆mµ = δ1e1(e1(µ)) + δ2e2(e2(µ)) + δ3e3(e3(µ)) + δ2e2(µ)l1 − δ1e1(µ)l2
−δ1e1(µ)λ − δ2e2(µ)µ.
[ ]
2 −∆Mλ− δ( ) = 2l1e1(µ)− δ2e1(µl1)− δ2l2e2(µ)− δ2e2(µl2)τ π δ { }1 2 2 2 − B ε+δ2λµl1 + δ2µ l2 + λ δ2l1 + δ 11l2 δ1δ2K
[ ]
−∆Mµ+ δ1l1e1(λ) + δ1e1(λl1) + δ1l2e2(λ) + δ1e2(λl+ 2)δ2 { }−δ 2 2 2 B ε2,1λµl2 − δ1λ l1 + µ δ2l1 + δ1l2 − δ1δ2K
which completes the proof.
When the integrability function µ = 0 we have the following corollary.
′
Corollary 1. Let π : (M3r , g) → (B2s , g ) be a pseudo-Riemannian submersion with an
adapted frame {e1, e2, e3} and the integrability functions l1, l2, λ, µ and σ with µ = 0 .
Then, the pseudo-Riemannian submersion π is biharmonic if and only if
{ }
(3.6) −δ ∆M1 λ+ λ δ1δ 22l1 + l22 − δ KB2 = 0,
δ1δ2l1e1(λ) + δ1δ2e1(λl1) + δ1δ2l2e2(λ) + δ1δ2e2(λl2)− δ1δ 22λ l1 = 0.

The following lemmas will be used to prove Classification Theorem.
′
Lemma 4. Let π : M3 2r (c) → (Bs , g ) be a pseudo-Riemannian submersion from a space
form of constant sectional curvature c. Then, for any orthonormal frame {e1, e2, e3}
on M3r (c) adapted to the pseudo-Riemannian submersion with e3 being vertical, all the
integrability functions l1, l2, λ, µ and σ are constant along fibers of π, i.e.,
(3.7) e3(l1) = e3(l2) = e3(µ) = e3(λ) = e3(σ) = 0
BIHARMONIC PSEUDO-RIEMANNIAN SUBMERSIONS FROM 3-MANIFOLDS 7
Proof. From definition, li = Fi ◦π for i = 1, 2 we can conclude that l1 and l2 are constant
along the fibers. It remains to show that
(3.8) e3(µ) = e3(λ) = e3(σ) = 0.
Using the Jacobi identity to the frame {e1, e2, e3}, we have
(3.9) 2e3(σ) + λl1 + µl2 + e2(λ)− e1(µ) = 0.
By using (3.9) and the fact that M31 (c) has constant sectional curvature c, calculating
RM1312, R
M
1313, R
M M
1323, R1212, R
M
1223, R
M M
2313, R2323 respectively, we get
i)e1(σ)− 2λσ = 0,
ii) δ1e1(λ) + δ1δ2δ3σ
2 − δ1λ2 + δ2µl1 = c,
iii)− e1(µ) + e3(σ) + λl1 + λµ = 0,
(3.10) iv)− δ2e2(l ) + δ e (l )− δ l21 1 1 2 2 1 − δ1l22 − 3δ 21δ2δ3σ = c,
v)e2(σ)− 2µσ = 0,
vi)− e2(λ)− e3(σ)− µl2 + λµ = 0,
vii) δ1δ2δ σ
2
3 + δ2e2(µ)− δ 21λl2 − δ2µ = c.
Applying e3 to both sides of the equation iv) of (3.10) and using e3e1 = [e3, e1] + e1e3
and e3e2 = [e3, e2] + e2e3, we obtain
σe3(σ) = 0,
which implies
e3(σ) = 0.
Using the last equation and applying e3 to both sides of the equations i) and v) of (3.10)
respectively, we get
e3(λ) = 0, e3(µ) = 0.

Case 1. Spacelike Fiber
Submersion
Signature of g New Orthonormal frame of Base Manifold
Signature of g′
π : (M31 , g) → 2 ′
′ ′
(B1 , g ) ε √ λ̄ √ µ̄ √ µ̄ √ λ̄ 2 21 = − ε1 + ε2, ε2 = − ε1 + ε ;if λ̄ − µ̄ > 02 2 2 2 2 2
(e , e , e ; +,−,+) λ̄ −µ̄ λ̄ −µ̄2 λ̄ −µ̄2 λ̄ −µ̄21 2 3 ′ √ µ̄ λ̄ ′ λ̄ µ̄ 2 2
(ε , ε ; +,−) ε1 = − ε2 1 + √ ε2, ε2 = −√ ε + √ ε ;if µ̄ − λ̄ > 01 2 2 2 2 2 2 1 2 2µ̄ −λ̄ µ̄ −λ̄ µ̄ −λ̄ µ̄2−λ̄
π : (M32 , g) → (B2 ′2 , g )
′ ′
(e1, e2, e3;−,−,+) ε = √ λ̄1 ε1 + √ µ̄ ε , ε = √ µ̄ ε − √ λ̄ ε2+ 2 2 2 2 2 1+ 2 + 2 2 2λ̄ µ̄ λ̄ µ̄ λ̄ µ̄ λ̄ +µ̄2
(ε1, ε2;−,−)
π : (M3, g) → (B2, g′)
′ ′
(e1, e
λ̄ µ̄ µ̄ λ̄
2, e3; +,+,+) ε1 = √ ε2 1 + √ ε2, ε √2 2 = − ε2 1 + √ ε+ 2 2λ̄ µ̄2 λ̄ +µ̄2 λ̄ +µ̄2 λ̄ +µ̄2
(ε1, ε2; +,+)
Table 1
Case 2. Timelike Fiber
8 İREM KÜPELI ERKEN AND CENGIZHAN MURATHAN
Submersion
Signature of g New Orthonormal frame of Base Manifold
Signature of g′
π : (M3 21 , g) → (B , g′)
′ ′
(e1, e2, e3; +,+,−) ε λ̄ µ̄ µ̄ λ̄1 = √ ε2 1 + √ ε2 2, ε2 = √ ε − √ ε+ 2 + 2 2 1 2 2λ̄ µ̄ λ̄ µ̄ λ̄ +µ̄2 λ̄ +µ̄2
(ε1, ε2; +,+)
′ ′ 2
π : (M32 , g) → (B2, g′) ε = −√ λ̄ ε + √ µ̄ ε µ̄1 1 1 2, ε2 = −√ ε1 + √ λ̄ ε2;if λ̄ − µ̄2 > 02 2 2 2 2 2 2 2
(e1, e2, e3; +−,−) λ̄ −µ̄ λ̄ −µ̄ λ̄ −µ̄ λ̄ −µ̄′ √ µ̄ ′ 2
(ε , ε : +,−) ε1 = − ε + √
λ̄
1 ε2, ε2 = −√ λ̄ ε1 + √ µ̄ ε2;if µ̄2 − λ̄ > 01 2 2 2 2 2µ̄2−λ̄ µ̄2−λ̄ µ̄2−λ̄ µ̄2−λ̄
π : (M33 , g) → (B22 , g′)
′ ′
(e1, e2, e3;−,−,−) ε = √ λ̄ ε µ̄ µ̄ λ̄1 1 + √ ε2, ε2 = √ ε1 − √ ε22
− − λ̄ + 2
2+ 2 2µ̄ λ̄ µ̄2 λ̄ +µ̄2 λ̄ +µ̄2
(ε1, ε2 : , )
Table 2
′
Lemma 5. Let π : (M3r (c), g) → (B2s , g ) be a pseudo-Riemannian submersion with an
adapted frame {e1, e2, e3} and the integrability functions l1, l2, λ, µ and σ . Then, there
{ }
′ ′ ′
exists another adapted orthonormal frame e1, e2, e3 = e3 on M
3
r (c) with integrability
′ ′
functions µ = 0, and σ = σ.
Proof. Applying the same method in ([19], Lemma 3.2) and using Lemma 4 , Table 1
and Table 2, one can complete the proof of the lemma. 
Now we will give a classification of biharmonic pseudo-Riemannian submersions.
Classification Theorem: Let π : M3 2r (c) → Bs be a pseudo-Riemannian submersion
from a space form of constant sectional curvature c. Then, π is biharmonic if and only
if it is equivalent to one of the following submersions:
Timelike Fiber Spacelike Fiber
π1 : H
3
3 (−1) → H22 (−4) = CH11 ; π6 : E32 → E22 ;
π2 : E
3
3 → E22 ; π7 : H31 (−1) → H21 (−4) = AH1;
π : H33 1 (−1) → H2(−4) = CH1; π : E3 → E28 1 1 ;
( )
π4 : E
3 → E21 ; π9 : S3(1) → S2 12 = CP 1;is proved by [19]
π : E35 2 → E21 ; π10 : E3 → E2,is proved by [19]
Table 3
Proof. By Lemma 5, we can choose an orthonormal frame {e1, e2, e3} adapted to the
pseudo-Riemannian submersion with integrability functions l1, l2, λ, µ and σ with
µ = 0. According to this frame (3.10) reduces to
a1)e1(σ)− 2λσ = 0,
a )δ e (λ) + δ δ δ σ2 − δ λ22 1 1 1 2 3 1 = c,
a3)λl1 = 0,
(3.11) a4)− δ2e2(l1) + δ e (l )− δ l21 1 2 2 1 − δ 21l2 − 3δ1δ δ σ22 3 = c,
a5)e2(σ) = 0,
a6)e2(λ) = 0,
a7)δ1δ
2
2δ3σ − δ1λl2 = c.
From a3) of (3.11), we have either λ = 0 or l1 = 0. If λ = 0, from (3.5) the tension field
of π vanishes. This means that pseudo-Riemannian submersion is harmonic. If l1 = 0
and λ =6 0, this case can not happen. We will prove this by a contradiction.
BIHARMONIC PSEUDO-RIEMANNIAN SUBMERSIONS FROM 3-MANIFOLDS 9
Case I: λ 6= 0, l1 = 0 and l2 = 0. So, from a4), a7) in (3.11), we have σ = c = 0. If
we put l1 = l2 = σ = 0 and µ = 0 into (3.6) we obtain
∆Mλ = 0,
which, one can easily get by using a2), a6) of (3.11) ,
λ3 = 0.
It follows that λ = 0 which is a contradiction.
Case II: λ 6= 0, l1 = 0 and l2 6= 0. In this case, by using l1 = 0 and a5), a6) and
a7) of (3.11), (3.6) reduces to
[ ]
(3.12) − δ1∆Mλ+ λ −δ2c− 3δ 2 21δ3σ + l2 = 0,
where KB = c + 3δ 21δ2δ3σ obtained from curvature formula for a pseudo-Riemannian
submersion. Using a1), a2) of (3.11) and after a straightforward calculation yields
∆Mλ = δ1e1(e1(λ)) − δ1e1(λ)l2 − δ1e1(λ)λ
∆Mλ = −5δ1δ2δ3λσ2 + δ λ31 + λc+ l2(−c+ δ 2 21δ2δ3σ − δ1λ ).
Substituting this into (3.12) and using a7) we obtain
[ ]
(3.13) λ δ3(6δ
2 2
2 − 3δ1)σ − λ − (2δ1 + δ2)c = 0.
We accept λ =6 0, so (3.13) is equivalent to
(3.14) λ2 = δ3(6δ2 − 3δ1)σ2 − (2δ1 + δ2)c.
After applying e1 to both sides of (3.14), we get
λe1(λ) = δ3(6δ2 − 3δ1)σe1(σ).
Combining this and a1) , a2) in (3.11), we have
λ(λ2 − δ 2 22δ3σ + δ1c) = 2δ3(6δ2 − 3δ1)λσ .
By assumption λ =6 0, this turned into
λ2 + δ1c = δ3(13δ2 − 6δ1)σ2,
or
(3.15) λ2 = δ 23(13δ2 − 6δ1)σ − δ1c.
Applying e1 to both sides of (3.15) and again using a1), a2) in (3.11) we get
(3.16) λ2 = δ3(27δ2 − 12δ1)σ2 − δ1c.
Combining (3.14), (3.15) with (3.16) we have λ = σ = c = 0. This implies there is a
contradiction. Because our assumption is λ =6 0.So we have λ = µ = 0. If we use (3.4) in
the first equation of (2.1) we get T (ei, ej) = 0, 1 ≤ i, j ≤ 3. It means that fiber is totally
geodesic. By (a2)of (3.11), we have
(3.17) δ1δ
2
2δ3σ = c.
Using the last equation and Theorem 3 , we get our classification. 
10 İREM KÜPELI ERKEN AND CENGIZHAN MURATHAN
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Faculty of Natural Sciences, Architecture and Engineering, Department of Mathematics,
Bursa Technical University, Bursa, TURKEY
E-mail address: irem.erken@btu.edu.tr
Art and Science Faculty,Department of Mathematics, Uludag University, 16059 Bursa,
TURKEY
E-mail address: cengiz@uludag.edu.tr