Bull. Math. Soc. Sci. Math. Roumanie
Tome 54(102) No. 1, 2011, 93–105
Slant submersions from almost Hermitian manifolds
by
Bayram S.ahin
Abstract
As a generalization of almost Hermitian submersions and anti-invariant
submersions, we introduce slant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the
geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also
find necessary and sufficient conditions for a slant submersion to be totally geodesic. Moreover, we obtain a decomposition theorem for the total
manifold of such submersions.
Key Words: Riemannian submersion, Hermitian manifold, almost
Hermitian submersion, anti-invariant Riemannian submersion, slant
submersion.
2010 Mathematics Subject Classification: Primary 53C15; Secondary 53B20,53C43.
1
Introduction
Let M̄ be an almost Hermitian manifold with complex structure J and M a Riemannian manifold isometrically immersed in M̄ . We note that submanifolds of
a Kähler manifold are determined by the behavior of the tangent bundle of the
submanifold under the action of the complex structure of the ambient manifold.
A submanifold M is called holomorphic (complex) if J(Tp M ) ⊂ Tp M , for every
p ∈ M , where Tp M denotes the tangent space to M at the point p. M is called
totally real if J(Tp M ) ⊂ Tp M ⊥ for every p ∈ M, where Tp M ⊥ denotes the normal
space to M at the point p. As a generalization of holomorphic and totally real
submanifolds, slant submanifolds were introduced by Chen in [5]. We recall that
the submanifold M is called slant [5] if for any p ∈ M and any X ∈ Tp M , the
angle between JX and Tp M is a constant θ(X) ∈ [0, π2 ], i.e, it does not depend
on the choice of p ∈ M and X ∈ Tp M . It follows that invariant and totally real
immersions are slant immersions with slant angle θ = 0 and θ = π2 respectively.
93
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Slant submersions
A slant immersion which is neither invariant nor totally real is called a proper
slant immersion.
On the other hand, Riemannian submersions between Riemannian manifolds
were studied by O’Neill [13] and Gray [10]. Later such submersions were considered between manifolds with differentiable structures. As an analogue of holomorphic submanifolds, Watson defined almost Hermitian submersions between
almost Hermitian manifolds and he showed that the base manifold and each fiber
have the same kind of structure as the total space, in most cases [15]. We note
that almost Hermitian submersions have been extended to the almost contact
manifolds [6], locally conformal Kähler manifolds [12] and quaternion Kähler manifolds [11].
Let M be a complex m−dimensional almost Hermitian manifold with Hermitian metric gM and almost complex structure JM and N be a complex n−dimensional almost Hermitian manifold with Hermitian metric gN and almost complex
structure JN . A Riemannian submersion F : M −→ N is called an almost Hermitian submersion if F is an almost complex map, i.e., F∗ JM = JN F∗ . The main
result of this notion is that the vertical and horizontal distributions are JM −
invariant. On the other hand, Riemannian submersions from almost Hermitian
manifolds onto Riemannian manifolds have been studied by many authors under
the assumption that the vertical spaces of such submersions are invariant with
respect to the complex structure. For instance, Escobales [8] studied Riemannian
submersions from complex projective space onto a Riemannian manifold under
the assumption that the fibers are connected, complex, totally geodesic submanifolds. One can see that this assumption implies that the vertical distribution is
invariant.
Recently, we introduced anti-invariant Riemannian submersions from almost
Hermitian manifolds onto Riemannian manifolds and investigated the geometry of
such submersions [14]. We now recall the definition of anti-invariant Riemannian
submersions. Let M be a complex m− dimensional almost Hermitian manifold
with Hermitian metric gM and almost complex structure J and N a Riemannian
manifold with Riemannian metric gN . Suppose that there exists a Riemannian
submersion F : M −→ N such that (kerF∗ ) is anti-invariant with respect to J,
i.e., J(kerF∗ ) ⊆ (kerF∗ )⊥ . Then we say that F is an anti-invariant Riemannian
submersion. In this paper, as a generalization of Hermitian submersions and
anti-invariant submersions, we define and study slant submersions from almost
Hermitian manifolds onto Riemannian manifolds.
The paper is organized as follows: In section 2, we present the basic information needed for this paper. In section 3, we give definition of slant Riemannian
submersions, provide examples and give a sufficient condition for slant submersions to be harmonic. We also investigate the geometry of leaves of the dis-
Slant submersions
95
tributions and obtain a decomposition theorem. Finally we give necessary and
sufficient conditions for such submersions to be totally geodesic.
2
Preliminaries
In this section, we define almost Hermitian manifolds, recall the notion of Riemannian submersions between Riemannian manifolds and give a brief review of
basic facts of Riemannian submersions.
Let (M̄ , g) be an almost Hermitian manifold. This means [16] that M̄ admits
a tensor field J of type (1, 1) on M̄ such that, ∀X, Y ∈ Γ(T M̄ ), we have
J 2 = −I,
g(X, Y ) = g(JX, JY ).
(2.1)
An almost Hermitian manifold M̄ is called Kähler manifold if
¯ X J)Y = 0, ∀X, Y ∈ Γ(T M̄ ),
(∇
(2.2)
¯ is the Levi-Civita connection on M̄ . Let (M m , g ) and (N n , g ) be
where ∇
M
N
Riemannian manifolds, where dim(M ) = m, dim(N ) = n and m > n. A Riemannian submersion F : M −→ N is a map from M onto N satisfying the
following axioms:
(S1) F has maximal rank.
(S2) The differential F∗ preserves the lenghts of horizontal vectors.
For each q ∈ N , F −1 (q) is an (m − n) dimensional submanifold of M . The submanifolds F −1 (q) are called fibers. A vector field on M is called vertical if it is
always tangent to fibers. A vector field on M is called horizontal if it is always
orthogonal to fibers. A vector field X on M is called basic if X is horizontal
and F − related to a vector field X∗ on N , i.e., F∗ Xp = X∗F (p) for all p ∈ M .
Note that we denote the projection morphisms on the distributions (kerF∗ ) and
(kerf F∗ )⊥ by V and H, respectively.
We recall the following lemma from O’Neill [13].
Lemma 2.1 Let F : M −→ N be a Riemannian submersion between Riemannian
manifolds and X, Y be basic vector fields of M . Then
(a) gM (X, Y ) = gN (X∗ , Y∗ ) ◦ F ,
(b) the horizontal part [X, Y ]H of [X, Y ] is a basic vector field and corresponds
to [X∗ , Y∗ ], i.e., F∗ ([X, Y ]H ) = [X∗ , Y∗ ].
(c) [V, X] is vertical for any vector field V of (kerF∗ ).
M
N
(d) (∇X Y )H is the basic vector field corresponding to ∇X∗ Y∗ .
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Slant submersions
The geometry of Riemannian submersions is characterized by O’Neill’s tensors
T and A defined for vector fields E, F on M by
AE F = H∇HE VF + V∇HE HF
(2.3)
TE F = H∇VE VF + V∇VE HF,
(2.4)
where ∇ is the Levi-Civita connection of gM . It is easy to see that a Riemannian
submersion F : M −→ N has totally geodesic fibers if and only if T vanishes
identically. For any E ∈ Γ(T M ), TE and AE are skew-symmetric operators on
(Γ(T M ), g) reversing the horizontal and the vertical distributions. It is also easy
to see that T is vertical, TE = TVE and A is horizontal, A = AHE . We note that
the tensor fields T and A satisfy
TU W
AX Y
= TW U, ∀U, W ∈ Γ(kerF∗ )
1
= −AY X = V[X, Y ], ∀X, Y ∈ Γ((kerF∗ )⊥ ).
2
(2.5)
(2.6)
On the other hand, from (2.3) and (2.4) we have
∇V W
ˆV W
= TV W + ∇
(2.7)
∇V X
= H∇V X + TV X
(2.8)
∇X V
= AX V + V∇X V
(2.9)
∇X Y
= H∇X Y + AX Y
(2.10)
ˆ V W = V∇V W . If X is
for X, Y ∈ Γ((kerF∗ )⊥ ) and V, W ∈ Γ(kerF∗ ), where ∇
basic, then H∇V X = AX V .
Finally, we recall the notion of harmonic maps between Riemannian manifolds.
Let (M, gM ) and (N, gN ) be Riemannian manifolds and suppose that ϕ : M −→ N
is a smooth map between them. Then the differential ϕ∗ of ϕ can be viewed a
section of the bundle Hom(T M, ϕ−1 T N ) −→ M, where ϕ−1 T N is the pullback
bundle which has fibers (ϕ−1 T N )p = Tϕ(p) N, p ∈ M . Hom(T M, ϕ−1 T N ) has
a connection ∇ induced from the Levi-Civita connection ∇M and the pullback
connection ∇ϕ . Then the second fundamental form of ϕ is given by
M
(∇ϕ∗ )(X, Y ) = ∇ϕ
X ϕ∗ (Y ) − ϕ∗ (∇X Y )
(2.11)
for X, Y ∈ Γ(T M ). It is known that the second fundamental form is symmetric.
A smooth map ϕ : (M, gM ) −→ (N, gN ) is said to be harmonic if trace(∇ϕ∗ ) = 0.
On the other hand, the tension field of ϕ is the section τ (ϕ) of Γ(ϕ−1 T N ) defined
by
m
X
τ (ϕ) = divϕ∗ =
(∇ϕ∗ )(ei , ei ),
(2.12)
i=1
where {e1 , ..., em } is the orthonormal frame on M . Then it follows that ϕ is
harmonic if and only if τ (ϕ) = 0, for details, see [1].
Slant submersions
3
97
Slant Submersions
In this section, we define slant submersions from an almost Hermitian manifold
onto a Riemannian manifold by using the definition of a slant distribution given
in [3]. We give examples and check the harmonicity of slant submersions. Then
we investigate the geometry of leaves of distributions and obtain a decomposition
theorem for the total manifold. We also obtain a necessary and sufficient condition for such submersions to be totally geodesic map.
We first recall the definition of the slant distribution. Given a submanifold
M , isometrically immersed in an almost Hermitian manifold (M̄ , gM̄ , J), a differentiable distribution D on M is said to be a slant distribution if for any non-zero
vector X ∈ Dp ; p ∈ M , the angle between JX and the vector space Dp is
constant, that is, it is independent of the choice of p ∈ M and X ∈ Dp . This
constant angle is called the slant angle of the slant distribution D [3]. Inspiring
from the above definition we present the following notion.
Definition 3.1. Let F be a Riemannian submersion from an almost Hermitian
manifold (M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ). If for any non-zero
vector X ∈ (kerF∗p ); p ∈ M1 , the angle θ(X) between JX and the space (kerF∗p )
is a constant, i.e. it is independent of the choice of the point p ∈ M1 and choice
of the tangent vector X in (kerF∗p ), then we say that F is a slant submersion.
In this case, the angle θ is called the slant angle of the slant submersion.
It is known that the distribution (kerF∗ ) is integrable for a Riemannian submersion between Riemannian manifolds. In fact, its leaves are F −1 (q), q ∈ M1 ,
i.e., fibers. Thus it follows from above definition that the fibers of a slant submersion are slant submanifolds of M1 , for slant submanifolds, see: [4].
We first give some examples of slant submersions.
Example 1. Every almost Hermitian submersion from an almost Hermitian manifold onto an almost Hermitian manifold is a slant submersion with θ = 0.
Example 2. Every anti-invariant Riemannian submersion from an almost Hermitian manifold to a Riemannian manifold is a slant submersion with θ = π2 .
A slant submersion is said to be proper if it is neither Hermitian nor antiinvariant Riemannian submersion.
Example 3. Consider the following Riemannian submersion given by
F :
R4
−→
(x1 , x2 , x3 , x4 )
R2
(x1 sin α − x3 cos α, x4 ).
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Slant submersions
Then for any 0 < α <
π
2,
F is a slant submersion with slant angle α.
Example 4. Consider the following Riemannian submersion given by
F :
R4
−→
(x1 , x2 , x3 , x4 )
R2
4
, x2 ).
( x1√−x
2
Then F is a slant submersion with slant angle θ =
π
4.
Let F be a Riemannian submersion from an almost Hermitian manifold M1
with the structure (g1 , J) onto a Riemannian manifold (M2 , g2 ). Then for X ∈
Γ(kerF∗ ), we write
JX = φX + ωX,
(3.1)
where φX and ωX are vertical and horizontal parts of JX.
Γ((kerF∗ )⊥ ), we have
JZ = BZ + CZ,
Also for Z ∈
(3.2)
where BZ and CZ are vertical and horizontal components of JZ. Using (2.7),
(2.8), (3.1) and (3.2) we obtain
(∇X ω)Y
= CTX Y − TX φY
(3.3)
(∇X φ)Y
= BTX Y − TX ωY,
(3.4)
where ∇ is the Levi-Civita connection on M1 and
(∇X ω)Y
=
(∇X φ)Y
=
ˆ XY
H∇X ωY − ω ∇
ˆ X φY − φ∇
ˆ XY
∇
for X, Y ∈ Γ(kerF∗ ). Let F be a proper slant submersion from an almost Hermitian manifold (M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ), then we say
that ω is parallel with respect to the Levi-Civita connection ∇ on (kerF∗ ) if its
covariant derivative with respect to ∇ vanishes, i.e., we have
(∇X ω)Y = ∇X ωY − ω(∇X Y ) = 0
for X, Y ∈ Γ(kerF∗ ).
The proof of the following result is exactly same with slant immersions (see
[4] or [2] for Sasakian case), therefore we omit its proof.
Theorem 3.1. Let F be a Riemannian submersion from an almost Hermitian
manifold (M1 , g1 , J) onto a Riemannian manifold (M2 , g2 ). Then F is a proper
slant submersion if and only if there exists a constant λ ∈ [−1, 0] such that
φ2 X = λX
Slant submersions
99
for X ∈ Γ(kerF∗ ). If F is a proper slant submersion, then λ = − cos2 θ.
By using above theorem, it is easy to see the following lemma.
Lemma 3.1. Let F be a proper slant submersion from an almost Hermitian
manifold (M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ) with slant angle θ.
Then, for any X, Y ∈ Γ(kerF∗ ), we have
g1 (φX, φY )
g1 (ωX, ωY )
=
=
cos2 θg1 (X, Y ),
2
sin θg1 (X, Y ).
(3.5)
(3.6)
We now denote the orthogonal complementary distribution to ω(kerF∗ ) in
(kerF∗ )⊥ by µ. Then we have the following.
Proposition 3.1. Let F be a proper slant submersion from an almost Hermitian
manifold (M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ). Then µ is invariant
with respect to J1 .
Proof: For V ∈ Γ(µ), using (2.1), we have
g1 (J1 V, ωX)
=
g1 (J1 V, J1 X) − g1 (J1 V, φX)
=
−g1 (J1 V, φX)
for X ∈ Γ(kerF∗ ). Then, using Theorem 3.1, we get
g1 (J1 V, ωX)
= g1 (V, J1 φX)
= g1 (V, φ2 X) + g1 (V, ωφX)
= − cos2 θg1 (V, X) + g1 (V, ωφX)
= g1 (V, ωφX) = 0
due to µ is orthogonal to ω(kerF∗ ). In a similar way, we have g1 (J1 V, Y ) =
−g1 (V, J1 Y ) = 0 due to J1 Y ∈ Γ((kerF∗ ) ⊕ ω(kerF∗ )) for V ∈ Γ(µ) and Y ∈
Γ(kerF∗ ). Thus proof is complete.
Corollary 3.1. Let F be a proper slant submersion from an almost Hermitian
manifold (M1m , g1 , J1 ) onto a Riemannian manifold (M2n , g2 ). Let
{e1 , ..., em−n }
be a local orthonormal basis of (kerF∗ ), then {csc ωe1 , ..., csc ωem−n } is a local
orthonormal basis of ω(kerF∗ ).
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Slant submersions
Proof: It will be enough to show that g1 (csc θωei , csc θωej ) = δij , for any i, j ∈
{1, ..., m−n
2 }. By using (3.6), we have
g1 (csc θωei , csc θωej ) = csc2 θ sin2 θg1 (ei , ej ) = δij ,
which proves the assertion.
We note that above Proposition 3.1 tells that the distributions µ and (kerF∗ )⊕
ω(kerF∗ ) are even dimensional. In fact it implies that the distribution (kerF∗ )
is even dimensional. More precisely, we have the following result whose proof is
similar to the above corollary.
Lemma 3.2. Let F be a proper slant submersion from an almost Hermitian
manifold (M1m , g1 , J1 ) onto a Riemannian manifold (M2n , g2 ). If e1 , ..., e m−n are
2
orthogonal unit vector fields in (kerF∗ ), then
{e1 , sec θφe1 , e2 , sec θφe2 , ..., e m−n , sec θφe m−n }
2
2
is a local orthonormal basis of (kerF∗ ).
Let F be a proper slant submersion from an almost Hermitian manifold
(M12n , J1 , g1 ) onto a Riemannian manifold (M2n , g2 ). As in slant immersions,
we call such an orthonormal frame
{e1 , sec θφe1 , e2 , sec θφe2 , ..., en , sec θφen , csc θωe1 , csc θωe2 , ..., csc θωen }
an adapted slant frame for slant submersions.
In the sequel, we show that the slant submersion puts some restrictions on
the dimensions of the distributions and the base manifold.
Proposition 3.2. Let F be a proper slant submersion from an almost Hermitian
manifold (M1m , g1 , J1 ) onto a Riemannian manifold (M2n , g2 ). Then dim(µ) =
2n − m. If µ = {0}, then n = m
2.
Proof: First note that dim(kerF∗ ) = m − n. Thus using Corollary 3.1, we have
dim((kerF∗ ) ⊕ ω(kerF∗ )) = 2(m − n). Since M1 is m− dimensional, we get
dim(µ) = 2n − m. Second assertion is clear.
We now check the harmonicity of slant submersions. But we first give a
preparatory lemma.
Slant submersions
101
Lemma 3.3. Let F be a proper slant submersion from a Kähler manifold onto
a Riemannian manifold. If ω is parallel with respect to ∇ on (kerF∗ ), then we
have
TφX φX = − cos2 θTX X
(3.7)
for X ∈ Γ(kerF∗ ).
Proof: If ω is parallel, then from (3.3) we have CTX Y = TX φY for X, Y ∈
Γ(kerF∗ ). Interchanging the role of X and Y , we get CTY X = TY φX. Thus we
have
CTX Y − CTY X = TX φY − TY φX.
Using (2.5) we derive
TX φY = TY φX.
(3.8)
2
Then substituting Y by φX we get TX φ X = TφX φX. Finally using Theorem
3.1 we obtain (3.7).
We now give a sufficient condition for a proper slant submersion to be harmonic.
Theorem 3.2. Let F be a proper slant submersion from a Kähler manifold onto
a Riemannian manifold. If ω is parallel with respect to ∇ on (kerF∗ ), then F is
a harmonic map.
Proof: Since
(∇F∗ )(Z1 , Z2 ) = 0
⊥
(3.9)
for Z1 , Z2 ∈ Γ((kerF∗ ) ).PA proper slant submersion F is harmonic if and only if
P
m−n
m−n
m−n
i=1 (∇F∗ )(ẽi , ẽi ) = −
i=1 F∗ (Tẽi ẽi ) = 0, where {ẽi }i=1 is an orthonormal
basis of (kerF∗ ). Thus using Lemma 3.2, we can write
m−n
2
τ =−
X
F∗ (Tei ei + Tsec θφei sec θφei ).
i=1
Hence we have
m−n
2
τ = −(
X
F∗ (Tei ei + sec2 θTφei φei )).
i=1
Then using (3.7) we arrive at
m−n
2
τ = −(
X
i=1
which shows that F is harmonic.
F∗ (Tei ei − Tei ei )) = 0
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Slant submersions
We now investigate the geometry of the leaves of the distributions (kerF∗ )
and (kerF∗ )⊥ .
Theorem 3.3. Let F be a proper slant submersion from a Kähler manifold
(M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ). Then the distribution (kerF∗ )
defines a totally geodesic foliation on M1 if and only if
g1 (H∇X ωφY, Z) = g1 (H∇X ωY, CZ) + g1 (TX ωY, BZ)
for X, Y ∈ Γ(kerF∗ ) and Z ∈ Γ((kerF∗ )⊥ ).
Proof: For X, Y ∈ Γ(kerF∗ ) and Z ∈ Γ((kerF∗ )⊥ ), from (2.1) and (3.1) we have
g1 (∇X Y, Z) = g1 (∇X φY, JZ) + g1 (∇X ωY, JZ).
Using (2.1), (3.1) and (3.2) we get
g1 (∇X Y, Z)
=
−g1 (∇X φ2 Y, Z) − g1 (∇X ωφY, Z)
+
g1 (∇X ωY, BZ) + g1 (∇X ωY, CZ).
Then from (2.8) and Theorem 3.1 we obtain
g1 (∇X Y, Z)
=
cos2 θg1 (∇X Y, Z) − g1 (H∇X ωφY, Z)
+
g1 (TX ωY, BZ) + g1 (H∇X ωY, CZ).
Hence we have
sin2 θg1 (∇X Y, Z)
=
−g1 (H∇X ωφY, Z)
+
g1 (TX ωY, BZ) + g1 (H∇X ωY, CZ)
which proves assertion.
In a similar way we have the following.
Theorem 3.4. Let F be a proper slant submersion from a Kähler manifold
(M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ). Then the distribution (kerF∗ )⊥
defines a totally geodesic foliation on M1 if and only if
g1 (H∇Z1 Z2 , ωφX) = g1 (AZ1 BZ2 + H∇Z1 CZ2 , ωX)
for X ∈ Γ(kerF∗ ) and Z1 , Z2 ∈ Γ((kerF∗ )⊥ ).
From Theorem 3.3 and Theorem 3.4 we have the following result.
Slant submersions
103
Corollary 3.1. Let F be a proper slant submersion from a Kähler manifold
(M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ). Then M1 is a locally product
Riemannian manifold if and only if
g1 (H∇Z1 Z2 , ωφX) = g1 (AZ1 BZ2 + H∇Z1 CZ2 , ωX)
and
g1 (H∇X ωφY, Z1 ) = g1 (H∇X ωY, CZ1 ) + g1 (TX ωY, BZ1 )
for X, Y, ∈ Γ(kerF∗ ) and Z1 , Z2 ∈ Γ((kerF∗ )⊥ ).
Finally we give necessary and sufficient conditions for a proper slant submersion to be totally geodesic. Recall that a differentiable map F between
Riemannian manifolds (M1 , g1 ) and (M2 , g2 ) is called a totally geodesic map
if (∇F∗ )(X, Y ) = 0 for all X, Y ∈ Γ(T M1 ).
Theorem 3.5. Let F be a proper slant submersion from a Kähler manifold
(M1 , g1 , J1 ) onto a Riemannian manifold (M2 , g2 ). Then F is totally geodesic
if and only if
g1 (TX ωY, BZ1 ) + g1 (H∇X ωY, CZ1 ) = g1 (H∇X ωφY, Z1 )
and
g1 (AZ1 BZ2 + H∇Z1 CZ2 , ωX) = −g1 (H∇Z1 ωφX, Z2 )
for Z1 , Z2 ∈ Γ((kerF∗ )⊥ ) and X, Y ∈ Γ(kerF∗ ).
Proof: First of all, since F is a Riemannian submersion we have
(∇F∗ )(Z1 , Z2 ) = 0
for Z1 , Z2 ∈ Γ((kerF∗ )⊥ ). Thus it is enough to show that (∇F∗ )(X, Y ) = 0
and (∇F∗ )(X, Z) = 0 for X, Y ∈ Γ(kerF∗ ) and Z ∈ Γ((kerF∗ )⊥ ). For X, Y ∈
Γ(kerF∗ ) and Z1 ∈ Γ((kerF∗ )⊥ ), since F is a Riemannian submersion, from (2.1),
(3.1) and (3.2) we have
g2 ((∇F∗ )(X, Y ), F∗ Z) = g1 (∇X JφY, Z) − g1 (∇X ωY, JZ).
Using again (3.1) and (3.2) we get
g2 ((∇F∗ )(X, Y ), F∗ Z)
=
g1 (∇X φ2 Y, Z) + g1 (∇X ωφY, Z)
−
g1 (∇X ωY, BZ) − g1 (∇X ωY, CZ).
Then Theorem 3.1, (2.7) and (2.8) imply that
g2 ((∇F∗ )(X, Y ), F∗ Z)
=
− cos2 θg1 (∇X Y, Z) + g1 (∇X ωφY, Z)
−
g1 (TX ωY, BZ) − g1 (H∇X ωY, CZ).
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Slant submersions
Hence we obtain
sin2 θg2 ((∇F∗ )(X, Y ), F∗ Z)
= g1 (∇X ωφY, Z) − g1 (TX ωY, BZ)
− g1 (H∇X ωY, CZ).
(3.10)
In a similar way, we get
sin2 θg2 ((∇F∗ )(X, Z1 ), F∗ (Z2 ))
= g1 (AZ1 BZ2 + H∇Z1 CZ2 , ωX)
+
g1 (H∇Z1 ωφX, Z2 ).
(3.11)
Then proof follows from (3.10) and (3.11).
Acknowledgements. The author is grateful to the referee for his/her valuable
comments and suggestions.
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Received: 10.05.2010
Revised: 10.09.2010
Accepted: 02.01.2011.
Inonu University, Department of Mathematics,
44280, Malatya-Turkey.
E-mail: [email protected]