Conformal anti-invariant $\xi^\perp-$submersions

0.24 MB·

by  Mehmet Akif Akyol

CONFORMAL ANTI-INVARIANT ξ⊥−SUBMERSIONS
7 MEHMET AKIF AKYOL AND YILMAZ GU¨NDU¨ZALP
1
0
2 Abstract. As a generalization of anti-invariant ξ⊥−Riemannian submersions,
n we introduce conformal anti-invariant ξ⊥−submersions from almost contact met-
a ric manifolds onto Riemannian manifolds. We investigate the geometry of foli-
J
ations which are arisen from the definition of a conformal submersion and find

7 necessary and sufficient conditions for a conformal anti-invariant ξ⊥−submersion
1
to be totally geodesic and harmonic, respectively. Moreover, we show that there

] are certain product structures on the total space of a conformal anti-invariant
G ξ⊥−submersion.
D
.
h
t
a 1. Introduction
m
[ Riemannian submersions between Riemannian manifolds were studied by O’Neill

[28] and Gray [22], for recent developments on the geometry of Riemannian sub-
1
v manifolds and Riemannian submersions, see:[9] and [15], respectively. In [43], the
7 Riemannian submersions were considered between almost Hermitian manifolds by
1
WatsonunderthenameofalmostHermitiansubmersions. Inthiscase, theRiemann-
1
5 iansubmersion isalso analmost complex mapping andconsequently thevertical and
0
horizontal distribution are invariant with respect to the almost complex structure of
.
1
the total manifold of the submersion. The study of anti-invariant Riemannian sub-
0
7 mersions from almost Hermitian manifolds were initiated by S¸ahin [38]. In this case,
1 the fibres are anti-invariant with respect to the almost complex structure of the to-
:
v tal manifold. Beside there are many notions related with anti-invariant Riemannian
i
X submersion (see: [2], [7], [8], [16], [19], [20], [21], [25], [30], [31], [32], [33], [35], [39],
r [40], [42]). In [13], Chinea defined almost contact Riemannian submersions between
a
almost contact metric manifolds and examined the differential geometric proper-
ties of Riemannian submersions between almost contact metric manifolds. More
′ ′ ′
precisely, let (M ,φ,ξ,η,g ) and (M ,φ ,ξ ,η ,g ) be almost contact manifolds with
1 1 2 2
dimM = 2m+1 and dimM = 2n+1. A Riemannian submersion π : M −→ M is
1 2 1 2
called the almost contact metric submersion if π is an almost contact mapping, i.e.,
′
φ π = π φ. An immediate consequence of the above definition is that the vertical
∗ ∗
andhorizontal distributionsareφ-invariant. Moreover, thecharacteristic vector field
ξ is horizontal. We note that only φ-holomorphic submersions have been considered
on almost contact manifolds [13].
One the other hand, as a generalization of Riemannian submersion, horizontally
conformal submersions are defined as follows [6]: Suppose that (M,g ) and (B,g )
M B
2010 Mathematics Subject Classification. 53C15,53C40.
Keywords andphrases. Almostcontactmetricmanifold,conformalsubmersion,conformalanti-
invariant ξ⊥−submersion.
1
2 MEHMET AKIFAKYOLAND YILMAZGU¨NDU¨ZALP
areRiemannian manifolds and π : M −→ B is a smooth submersion, then π iscalled
a horizontally conformal submersion, if there is a positive function λ such that
λ2g (X,Y) = g (π X,π Y)
M B ∗ ∗
for every X,Y ∈ Γ((kerπ )⊥). It is obvious that every Riemannian submersion is
∗
a particular horizontally conformal submersion with λ = 1. We note that hori-
zontally conformal submersions are special horizontally conformal maps which were
introduced independently by Fuglede[14] and Ishihara [23]. We also note that a hor-
izontally conformal submersion π : M −→ B is said to be horizontally homothetic
if the gradient of its dilation λ is vertical, i.e.,
H(gradλ) = 0 (1.1)
atp ∈ M, whereH istheprojectiononthehorizontal space(kerπ )⊥. Forconformal
∗
submersion, see: [6], [17], [29].
As a generalization of holomorphic submersions, conformal holomorphic submer-
sions were studied by Gudmundsson and Wood [18]. They obtained necessary and
sufficient conditions for con- formal holomorphic submersions to be a harmonic mor-
phism, see also [10], [11] and [12] for the harmonicity of conformal holomorphic
submersions.
Recently, in [3] we have introduced conformal anti-invariant submersions from
almost Hermitian manifolds onto Riemannian manifolds and investigated the ge-
ometry of such submersions. (See also:[1]) We showed that the geometry of such
submersions are different from anti-invariant Riemannian submersions. In this pa-
per, we consider conformal anti-invariant ξ⊥−submersions from an almost contact
metric manifold under the assumption that the fibers are anti-invariant with respect
to the tensor field of type (1,1) of the almost contact manifold.
The paper is organized as follows. In the second section, we gather main notions
and formulas for other sections. In section 3, we introduce conformal anti-invariant
ξ⊥−submersions from almost contact metric manifolds onto Riemannian manifolds,
investigates thegeometry ofleaves ofthe horizontal distribution andthe vertical dis-
tribution and find necessary and sufficient conditions for a conformal anti-invariant
ξ⊥−submersion to be totally geodesic and harmonic, respectively. In section 4, we
show that there are certain product structures on the total space of a conformal
anti-invariant ξ⊥−submersion.
2. Preliminaries
In this section, we define almost contact metric manifolds, recall the notion of
(horizontally) conformal submersions between Riemannian manifolds and give a
brief review of basic facts of (horizontally) conformal submersions.
Let(M,g )beanalmostcontactmetricmanifoldwithstructuretensors(φ,ξ,η,g )
M M
where φ is a tensor field of type (1,1), ξ is a vector field, η is a 1-form and g is the
M
Riemannian metric on M. Then these tensors satisfy [5]
φξ = 0, ηoφ = 0, η(ξ) = 1 (2.1)
φ2 = −I +η ⊗ξ and g (φX,φY) = g (X,Y)−η(X)η(Y), (2.2)
M M
CONFORMAL ANTI-INVARIANT ξ⊥−SUBMERSIONS 3
where I denotes the identity endomorphism of TM and X,Y are any vector fields
on M. Moreover, if M is Sasakian [37], then we have
(∇ φ)Y = −g (X,Y)ξ +η(Y)X and ∇ ξ = φX, (2.3)
X M X
where ∇ is the connection of Levi-Civita covariant differentiation.
Conformal submersions belong to a wide class of conformal maps that we are
going to recall their definition, but we will not study such maps in this paper.
Definition 2.1. ([6]) Let ϕ : (Mm,g) −→ (Nn,h) be a smooth map between Rie-
mannian manifolds, and let x ∈ M. Then ϕ is called horizontally weakly conformal
or semi conformal at x if either
(i) dϕ = 0, or
x
(ii) dϕ maps horizontal space H = (ker(dϕ ))⊥ conformally onto T N, i.e.,
x x x ϕ∗
dϕ is surjective and there exists a number Λ(x) 6= 0 such that
x
h(dϕ X,dϕ Y) = Λ(x)g(X,Y) (X,Y ∈ H ). (2.4)
x x x
A point x is of type (i) in Definition if and only if it is a critical point of ϕ;
we shall call a point of type (ii) a regular point. At a critical point, dϕ has rank
x
0; at a regular point, dϕ has rank n and ϕ is submersion. The number Λ(x) is
x
called the square dilation (of ϕ at x); it is necessarily non-negative; its square root
λ(x) = Λ(x) is called the dilation (of ϕ at x). The map ϕ is called horizontally
p
weakly conformal or semi conformal (on M) if it is horizontally weakly conformal
at every point of M. It is clear that if ϕ has no critical points, then we call it a
(horizontally) conformal submersion.
Next, we recall the following definition from [17]. Let π : M −→ N be a submer-
sion. A vector field E on M is said to be projectable if there exists a vector field Eˇ
on N, such that dπ(E ) = Eˇ for all x ∈ M. In this case E and Eˇ are called π−
x π(x)
related. A horizontal vector field Y on (M,g) is called basic, if it is projectable. It
is well known fact, that is Zˇ is a vector field on N, then there exists a unique basic
vector field Z on M, such that Z and Zˇ are π− related. The vector field Z is called
the horizontal lift of Zˇ.
The fundamental tensors of a submersion were introduced in [28]. They play a
similar role to that of the second fundamental form of an immersion. More precisely,
O’Neill’s tensors T and A defined for vector fields E,F on M by
A F = V∇ HF +H∇ VF (2.5)
E HE HE
T F = H∇ VF +V∇ HF (2.6)
E VE VE
where V and H are the vertical and horizontal projections (see [15]). On the other
hand, from (2.5) and (2.6), we have
∇ W = T W +∇ˆ W (2.7)
V V V
∇ X = H∇ X +T X (2.8)
V V V
∇ V = A V +V∇ V (2.9)
X X X
∇ Y = H∇ Y +A Y (2.10)
X X X
4 MEHMET AKIFAKYOLAND YILMAZGU¨NDU¨ZALP
for X,Y ∈ Γ((kerπ )⊥) and V,W ∈ Γ(kerπ ), where ∇ˆ W = V∇ W. If X is basic,
∗ ∗ V V
then H∇ X = A V. It is easily seen that for x ∈ M, X ∈ H and V the linear
V X x x
operators T , A : T M −→ T M are skew-symmetric, that is
V X X X
−g(T E,F) = g(E,T F) and −g(A E,F) = g(E,A F)
V V X X
for all E,F ∈ T M. We also see that the restriction of T to the vertical distribution
x
T | is exactly the second fundamental form of the fibres of π. Since T skew-
V×V V
symmetric we get: π has totally geodesic fibres if and only if T ≡ 0. For the special
case when π is horizontally conformal we have the following:
Proposition 2.1. ([17]) Let π : (Mm,g) −→ (Nn,h) be a horizontally conformal
submersion with dilation ∇ and X,Y be horizontal vectors, then
1 1
A Y = {V[X,Y]−λ2g(X,Y)grad ( )}. (2.11)
X V
2 λ2
We see that the skew-symmetric part of A |(kerπ∗)⊥×(kerπ∗)⊥ measures the obstruc-
tion integrability of the horizontal distribution (kerπ )⊥.
∗
Let(M,g )and(N,g )beRiemannianmanifoldsandsuppose thatπ : M −→ N
M N
is a smooth map between them. The differential π of π can be viewed a section of
∗
the bundle Hom(TM,π−1TN) −→ M, where π−1TN is the pullback bundle which
has fibres (π−1TN) = T N, p ∈ M. Hom(TM,π−1TN) has a connection ∇
p π(p)
induced from the Levi-Civita connection ∇M and the pullback connection. Then
the second fundamental form of π is given by
∇π : Γ(TM)×Γ(TM) −→ Γ(TN)
∗
defined by
(∇π )(X,Y) = ∇π π (Y)−π (∇MY) (2.12)
∗ X ∗ ∗ X
for X,Y ∈ Γ(TM), where ∇π is the pullback connection. It is known that the
second fundamental form is symmetric.
Lemma 2.1. [44] Let (M,g ) and (N,g ) be Riemannian manifolds and suppose
M N
that ϕ : M −→ N is a smooth map between them. Then we have
∇ϕϕ (Y)−∇ϕϕ (X)−ϕ ([X,Y]) = 0 (2.13)
X ∗ Y ∗ ∗
for X,Y ∈ Γ(TM).
A smooth map π : (M,g ) −→ (N,g ) is said to be harmonic if trace(∇π ) = 0.
M N ∗
On the other hand, the tension field of π is the section τ(π) of Γ(π−1TN) defined
by
m
τ(π) = divπ∗ = X(∇π∗)(ei,ei), (2.14)
i=1
where{e ,...,e }istheorthonormalframeonM. Then itfollowsthatπ isharmonic
1 m
if and only if τ(π) = 0 (for details, see [6]). Finally, we recall the following lemma
from [6].
Lemma 2.2. Suppose that π : M −→ N is a horizontally conformal submersion.
Then, for any horizontal vector fields X,Y and vertical fields V,W we have
CONFORMAL ANTI-INVARIANT ξ⊥−SUBMERSIONS 5
(i) (∇π )(X,Y) = X(lnλ)π Y +Y(lnλ)π X −g(X,Y)π (gradlnλ);
∗ ∗ ∗ ∗
(ii) (∇π )(V,W) = −π (T W);
∗ ∗ V
(iii) (∇π )(X,V) = −π (∇MV) = −π (A V).
∗ ∗ X ∗ X
3. Conformal Anti-invariant ξ⊥−submersions
In this section, we define conformal anti-invariant ξ⊥−submersions from an al-
most contact metric manifold onto a Riemannian manifold and investigate the in-
tegrability of distributions and obtain a necessary and sufficient condition for such
submersions to be totally geodesic map. We also investigate the harmonicity of such
submersions.
Definition 3.1. Let (M,φ,ξ,η,g ) be an almost contact metric manifold and and
M
(N,g ) be a Riemannian manifold. We suppose that there exist a horizontally con-
N
formal submersion π : M −→ N such that ξ is normal to kerπ and kerπ is
∗ ∗
anti-invariant with respect to φ, i.e., φ(kerπ ) ⊂ (kerπ )⊥. Then we say that π is a
∗ ∗
conformal anti-invariant ξ⊥−submersion.
Here, we assume that if π : (M,φ,ξ,η,g ) −→ (N,g ) is a conformal anti-
M N
invariant ξ⊥−submersion from a Sasakian manifold (M,φ,ξ,η,g ) to a Riemannian
M
manifold (N,g ). Then from Definition 3.1, we have φ(kerπ )⊥ ∩ kerπ 6= 0. We
N ∗ ∗
denote the complementary orthogonal distribution to φ(kerπ ) in (kerπ )⊥ by µ.
∗ ∗
Then we have
(kerπ )⊥ = φ(kerπ )⊕µ. (3.1)
∗ ∗
We can easily to see that µ is an invariant distribution of (kerπ )⊥, with respect to
∗
φ. Hence µ contains ξ. Thus, for X ∈ Γ((kerπ )⊥), we have
∗
φX = BX +CX, (3.2)
whereBX ∈ Γ(kerπ )andCX ∈ Γ(µ).Ontheother hand, since π ((kerπ )⊥) = TN
∗ ∗ ∗
and π is a conformal submersion, using (3.2) we derive 1 g (π φV,π CX) = 0 for
λ2 N ∗ ∗
any X ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ), which implies that
∗ ∗
TN = π (φkerπ )⊕π (µ). (3.3)
∗ ∗ ∗
Remark 3.1. We note that every anti-invariant ξ⊥−submersion from an almost
contactmanifoldonto a Riemannianmanifoldis aconformalanti-invariantξ⊥−submersion
with λ = I, where I denotes the identity function [24].
Lemma 3.1. Let π be a conformal anti-invariant ξ⊥-submersion from a Sasakian
manifold (M,φ,ξ,η,g ) onto a Riemannian manifold (N,g ). Then we have
M N
A ξ = −BX, (3.4)
X
T ξ = 0, (3.5)
V
g (CY,φV) = 0, (3.6)
M
g (∇ CY,φV) = −g (CY,φA V) (3.7)
M X M X
for X,Y,ξ ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ).
∗ ∗
6 MEHMET AKIFAKYOLAND YILMAZGU¨NDU¨ZALP
Proof. By virtue of (2.3), (2.10) and (3.2) we have (3.4). Using (2.3) and (2.8) we
get (3.5). By using (2.2), for Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ), we have
∗ ∗
g (CY,φV) = g (φY−BY,φV) = g (φY,φV) = g (Y,V)+η(Y)η(V) = g (Y,V) = 0,
M M M M M
since BY ∈ Γ(kerπ ) and φV,ξ ∈ Γ((kerπ )⊥). Differentiating (3.6) with respect to
∗ ∗
X, we get
g (∇ CY,φV) = −g (CY,∇ φV)
M X M X
= −g (CY,(∇ φ)V)−g (CY,φ(∇ V))
M X M X
= −g (CY,φ(∇ V))
M X
= −g (CY,φA V)−g (CY,φV∇ V)
M X M X
= −g (CY,φA V)
M X
due to φV∇ V ∈ Γ(φkerπ ). Our assertion is complete. (cid:3)
X ∗
Since the distribution kerπ is integrable, we only study the integrability of the
∗
distribution (kerπ )⊥ and then we investigate the geometry of leaves of kerπ and
∗ ∗
(kerπ )⊥.
∗
Theorem 3.1. Let π : (M,φ,ξ,η,g ) −→ (N,g ) be a conformal anti-invariant
M N
ξ⊥−submersion from a Sasakian manifold (M,φ,ξ,η,g ) onto a Riemannian man-
M
ifold (N,g ). Then the following assertions are equivalent to each other;
N
(a) (kerπ )⊥ is integrable,
∗
1
(b) g (∇ππ CX −∇π π CY,π φV) = g (A BY −A BX −CY(lnλ)X +CX(lnλ)Y
λ2 N Y ∗ X ∗ ∗ M X Y
−2g (CX,Y)lnλ−η(Y)X +η(X)Y,φV)
M
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ).
∗ ∗
Proof. From (2.2) and (2.3), we obtain
g (∇ Y,V) = g (∇ φY,φV)−η(Y)g (X,φV). (3.8)
M X M X M
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ). Then, from (3.2) and (3.8), we have
∗ ∗
g ([X,Y],V) = g (∇ φY,φV)−g (∇ φX,φV)−η(Y)g (X,φV)+η(X)g (Y,φV)
M M X M Y M M
= g (∇ BY,φV)+g (∇ CY,φV)−g (∇ BX,φV)−g (∇ CX,φV)
M X M X M Y M Y
−η(Y)g (X,φV)+η(X)g (Y,φV).
M M
Using (2.9) and if we take into account that π is a conformal submersion, we obtain
1
g ([X,Y],V) = g (A BY −A BX,φV)+ g (π (∇ CY),π φV)
M M X Y N ∗ X ∗
λ2
1
− g (π (∇ CX),π φV)−η(Y)g (X,φV)+η(X)g (Y,φV).
N ∗ Y ∗ M M
λ2
CONFORMAL ANTI-INVARIANT ξ⊥−SUBMERSIONS 7
Thus, from (2.12) and Lemma 2.2 we derive
g ([X,Y],V) = g (A BY −A BX,φV)−g (Hgradlnλ,X)g (CY,φV)
M M X Y M M
−g (Hgradlnλ,CY)g (X,φV)+g (X,CY)g (Hgradlnλ,φV)
M M M M
1
+ g (∇π π CY,π φV)+g (Hgradlnλ,Y)g (CX,φV)
λ2 N X ∗ ∗ M M
+g (Hgradlnλ,CX)g (Y,φV)−g (Y,CX)g (Hgradlnλ,φV)
M M M M
1
− g (∇ππ CX,π φV)−η(Y)g (X,φV)+η(X)g (Y,φV).
λ2 N Y ∗ ∗ M M
Moreover, using (3.6), we obtain
g ([X,Y],V) = g (A BY −A BX −CY(lnλ)X +CX(lnλ)Y −2g (CX,Y)lnλ
M M X Y M
1
−η(Y)X +η(X)Y,φV)− g (∇ππ CX −∇π π CY,π φV).
λ2 N Y ∗ X ∗ ∗
This show that (a) ⇔ (b). (cid:3)
From Theorem 3.1, we deduce the following which shows that a conformal anti-
invariant ξ⊥−submersion with integrable (kerπ )⊥ turns out to be a horizontally
∗
homothetic submersion.
Theorem 3.2. Let π be a conformal anti-invariant ξ⊥−submersion from a Sasakian
manifold (M,φ,ξ,η,g ) onto a Riemannian manifold (N,g ). Then any two con-
M N
ditions below imply the third;
(i) (kerπ )⊥ is integrable.
∗
(ii) π is horizontally homothetic submersion.
(iii) g (∇ππ CX −∇π π CY,π φV) = λ2g (A BY −A BX −η(Y)X +η(X)Y,φV)
N Y ∗ X ∗ ∗ M X Y
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ).
∗ ∗
Proof. From Theorem 3.1, we have
g ([X,Y],V) = g (A BY −A BX −CY(lnλ)X +CX(lnλ)Y −2g (CX,Y)lnλ
M M X Y M
1
−η(Y)X +η(X)Y,φV)− g (∇ππ CX −∇π π CY,π φV)
λ2 N Y ∗ X ∗ ∗
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ). Now, if we have (i) and (iii), then we
∗ ∗
arrive at
−g (Hgradlnλ,CY)g (X,φV)+g (Hgradlnλ,CX)g (Y,φV) (3.9)
M M M M
−2g (CX,Y)g (Hgradlnλ,φV) = 0.
M M
Now, taking Y = φV in (3.9) for V ∈ Γ(kerπ ), using (2.2) and (3.6), we get
∗
g (Hgradlnλ,CX)g (φV,φV)) = g (Hgradlnλ,CX){g (V,V)−η(V)η(V)}
M M M M
= g (Hgradlnλ,CX)g (V,V) = 0.
M M
8 MEHMET AKIFAKYOLAND YILMAZGU¨NDU¨ZALP
Hence λ is a constant on Γ(µ). On the other hand, taking Y = CX in (3.9) for
X ∈ Γ(µ) and using (3.6) we derive
−g (Hgradlnλ,C2Y)g (X,φV)+g (Hgradlnλ,CX)g (CX,φV)
M M M M
−2g (CX,CX)g (Hgradlnλ,φV) = 0,
M M
thus, we arrive at
2g (CX,CX)g (Hgradlnλ,φV) = 0.
M M
From above equation, λ is a constant on Γ(φkerπ ). Similarly, one can obtain the
∗
(cid:3)
other assertions.
Remark 3.2. We assume that (kerπ )⊥ = φkerπ ⊕{ξ}. Using (3.2) one can prove
∗ ∗
that CX = 0.
Hence we have the following corollary.
Corollary 3.1. Let π be a conformalanti-invariantξ⊥−submersion from a Sasakian
manifold (M,φ,ξ,η,g ) onto a Riemannian manifold (N,g ) with (kerπ )⊥ =
M N ∗
φ(kerπ )⊕ < ξ >. Then the following assertions are equivalent to each other;
∗
(i) (kerπ )⊥ is integrable
∗
(ii) A φY +η(X)Y = A φX +η(Y)X
X Y
(iii) (∇π )(X,φY)+η(Y)π X = (∇π )(Y,φX)+η(X)π Y
∗ ∗ ∗ ∗
for X,Y ∈ Γ((kerπ )⊥).
∗
For the geometry of leaves of the horizontal distribution, we have the following
theorem.
Theorem 3.3. Let π : (M,φ,ξ,η,g ) −→ (N,g ) be a conformal anti-invariant
M N
ξ⊥-submersion from a Sasakian manifold (M,φ,ξ,η,g ) onto a Riemannian mani-
M
fold (N,g ). Then the following assertions are equivalent to each other;
N
(i) (kerπ )⊥ defines a totally geodesic foliation on M.
∗
1
(ii) − g (∇π π CY,π φV) = g (A BY −CY(lnλ)X +g (X,CY)lnλ−η(Y)X,φV)
λ2 N X ∗ ∗ M X M
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ).
∗ ∗
Proof. By using (2.2), (2.9), (2.10), (3.1), (3.2) and (3.8), have
g (∇ Y,V) = g (A BY,φV)+g (∇ CY,φV)−η(Y)g (X,φV)
M X M X M X M
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ). Since π is a conformal submersion, using
∗ ∗
(2.12) and Lemma (2.2) we arrive at
1
g (∇ Y,V) = g (A BY,φV)− g (Hgradlnλ,X)g (π CY,π φV)
M X M X M N ∗ ∗
λ2
1
− g (Hgradlnλ,CY)g (π X,π φV)
M N ∗ ∗
λ2
1
+ g (X,CY)g (π (Hgradlnλ),π φV)
M N ∗ ∗
λ2
1
+ g (∇π π CY,π φV)−η(Y)g (X,φV).
λ2 N X ∗ ∗ M
CONFORMAL ANTI-INVARIANT ξ⊥−SUBMERSIONS 9
Moreover, using Definiton 3.1 and (3.6) we obtain
g (∇ Y,V) = g (A BY −CY(lnλ)X +g (X,CY)lnλ−η(Y)X,φV)
M X M X M
1
+ g (∇ π CY,π φV)
λ2 N π∗X ∗ ∗
which tells that (i) ⇔ (ii). (cid:3)
From Theorem 3.3, we also deduce the following characterization.
Theorem 3.4. Let π be a conformal anti-invariant ξ⊥−submersion from a Sasakian
manifold (M,φ,ξ,η,g ) onto a Riemannian manifold (N,g ). Then any two con-
M N
ditions below imply the third;
(i) (kerπ )⊥ defines a totally geodesic foliation on M.
∗
(ii) π is a horizontally homothetic submersion.
(iii) g (∇π π CY,π φV) = λ2g (−A BY +η(Y)X,φV)
N X ∗ ∗ M X
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ).
∗ ∗
Proof. For X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ), from Theorem 3.3, we have
∗ ∗
g (∇ Y,V) = g (A BY −CY(lnλ)X +g (X,CY)lnλ−η(Y)X,φV)
M X M X M
1
+ g (∇π π CY,π φV).
λ2 N X ∗ ∗
Now, if we have (i) and (iii), then we obtain
−g (Hgradlnλ,CY)g (X,φV)+g (Hgradlnλ,φV)g (X,CY) = 0. (3.10)
M M M M
Now, takingX = CY)in(3.10)andusing(3.6),wegetg (Hgradlnλ,φV)g (X,CY) =
M M
0. Hence, λ is a constant on Γ(φkerπ ). On the other hand, taking X = φV in (3.10)
∗
and using (3.6) we derive
g (Hgradlnλ,CY)g (φV,φV)) = g (Hgradlnλ,CY){g (V,V)−η(V)η(V)}
M M M M
= g (Hgradlnλ,CY)g (V,V) = 0.
M M
From above equation, λ is a constant on Γ(µ). Similarly, one can obtain the other
(cid:3)
assertions.
In particular, as an analogue of a conformal Lagrangian submersion in [3], we
have the following corollary.
Corollary 3.2. Let π be a conformalanti-invariantξ⊥−submersion from a Sasakian
manifold (M,φ,ξ,η,g ) onto a Riemannian manifold (N,g ) with (kerπ )⊥ =
M N ∗
φ(kerπ )⊕ < ξ >. Then the following assertions are equivalent to each other;
∗
(i) (kerπ )⊥ defines a totally geodesic foliation on M.
∗
(ii) A BY = η(Y)X
X
(iii) (∇π )(X,φV) = −η(Y)π X
∗ ∗
for X,Y ∈ Γ((kerπ )⊥) and V ∈ Γ(kerπ ).
∗ ∗
In the sequel we are going to investigate the geometry of leaves of the distribution
kerπ .
∗
10 MEHMET AKIFAKYOLAND YILMAZGU¨NDU¨ZALP
Theorem 3.5. Let π be a conformal anti-invariant ξ⊥−submersion from a Sasakian
manifold (M,φ,ξ,η,g ) onto a Riemannian manifold (N,g ). Then the following
M N
assertions are equivalent to each other;
(i) kerπ defines a totally geodesic foliation on M.
∗
1
(ii) − g (∇π π φV,π φCX) = g (φCX(lnλ)φV −T BX,φV)+η(∇ V)η(CX)
λ2 N φW ∗ ∗ M V φW
for V,W ∈ Γ(kerπ ) and X ∈ Γ((kerπ )⊥).
∗ ∗
Proof. Since g (W,ξ) = 0, using (2.3) we have g (∇ W,ξ) = −g (W,∇ ξ) =
M M V M V
−g (W,φV) = 0 for V,W ∈ Γ(kerπ ) and ξ ∈ Γ((kerπ )⊥). Thus we have
M ∗ ∗
g (∇ W,X) = g (φ∇ W,φX)+η(∇ W)η(X)
M V M V V
= g (φ∇ φW,φX)
M V
= g (∇ φW,φX)−g ((∇ φ)W,φX).
M V M V
Using (2.3), (2.7) and (3.2) we have
g (∇ W,X) = g (T φW,BX)+g (H∇ φW,CX).
M V M V M V
Since ∇ is torsion free and [V,φW] ∈ Γ(kerπ ) we obtain
∗
g (∇ W,X) = g (T φW,BX)+g (∇ V,CX).
M V M V M φW
Using (2.3) and (2.10) we have
g (∇ W,X) = g (T φW,BX)+g (φ∇ V,φCX)+η(∇ V)η(CX)
M V M V M φW φW
= g (T φW,BX)+g (∇ φV,φCX)+η(∇ V)η(CX)
M V M φW φW
here we have used that µ is invariant. Using (2.12) and Lemma 2.2 (i) and if we
take into account that π is a conformal submersion, we obtain
1
g (∇ V,X) = g (T φW,BX)+ g (Hgradlnλ,φW)g (π φV,π φCX)
M U M V M N ∗ ∗
λ2
1
− g (Hgradlnλ,φV)g (π φW,π φCX)
M N ∗ ∗
λ2
1
+g (φW,φV) g (π (Hgradlnλ),π φCX)
M N ∗ ∗
λ2
1
+ g (∇ π φV,π φCX)+η(∇ V)η(CX).
λ2 N π∗φW ∗ ∗ φW
Moreover, using Definition 3.1 and (3.6), we obtain
g (∇ V,X) = g (φCX(lnλ)φV −T BX,φV)+η(∇ V)η(CX)
M U M V φW
1
+ g (∇π π φV,π φCX)
λ2 N φW ∗ ∗
which tells that (i) ⇔ (ii). (cid:3)
From Theorem 3.5, we deduce the following result.
Theorem 3.6. Let π be a conformal anti-invariant ξ⊥−submersion from a Sasakian
manifold (M,φ,ξ,η,g ) onto a Riemannian manifold (N,g ). Then any two con-
M N
ditions below imply the third;