Conformal anti-invariantSubmersions from Sasakian manifolds
Sushil Kumar
Department of Mathematics and Astronomy,University of Lucknow, Lucknow-India.
Rajendra Prasad
Department of Mathematics and Astronomy,University of Lucknow, Lucknow-India.
Abstract
In this paper we define conformal anti-invariant submersions from almost contactmetric manifolds onto Riemannian manifolds. We obtain some results on conformalanti-invariant submersions from Sasakian manifolds onto Riemannian manifolds.We also give the necessary and sufficient conditions for a conformal anti-invariantsubmersions to be harmonic and totally geodesic. Moreover, we obtain decomposi-tion theorems by using the existence of conformal anti-invariant submersions fromSasakian manifolds onto Riemannian manifolds. Finally, we give some examplesof conformal anti-invariant submersions such that characteristic vector field ξ isvertical.
The theory of smooth maps between Riemannian manifolds has been extensively studiedin Riemannian geometry. Such maps are useful for comparing geometric structuresbetween two manifolds.
3578 Sushil Kumar and Rajendra Prasad
In 1966, O’Neill [19] initiated the study of Riemannian submersion between Rieman-nian manifolds. It was found beneficial if one should study such submersions betweenmanifolds with differentiable structures. When Watson was studying almost Hermitiansubmersions between almost Hermitian manifolds he found that the base manifold andeach fibre have the same kind of structure as the total space, in most cases [25]. Wenote that almost Hermitian submersions have been extended to the almost contact metricsubmersions [7], locally conformal Kahler submersions [16] etc.
We have so many submersions. Some of them are: semi-Riemannian submersionand Lorentzian submersion [8], semi-invariant submersion [23], slant submersion ([6],[22]), contact-complex submersion [12], anti-invariant Riemannian submersions fromCosymplectic manifold [18] etc. As we know that Riemannian submersions are relatedwith Physics and have their applications in the Yang-Mills theory [24], Kaluza-Kleintheory ([5], [13]), supergravity and superstring theories ([14], [17]). In 2010, Sahindefined anti-invariant Riemannian submersions from almost Hermitian manifolds ontoRiemannian manifolds [21] etc.
As a generalization of Riemannian submersions, horizontally conformal submersionare introduced as follows [2]. Let (M, gM) and (N, gN) be two Riemannian manifoldsof dimension m and n respectively. A smooth map f : (M, gM) → (N, gN) is called ahorizontally conformal submersion, if there is a positive function λ such that
λ2gM(U, V ) = gN(f∗U, f∗V ), (1.1)
for every U, V ∈ (ker f∗)⊥. It is evident that every Riemannian submersion is a particularhorizontally conformal submersion with λ = 1. Let f is a smooth map between givenRiemannian manifolds and x ∈ M . Then, f is called horizontally weakly conformal mapat x if either (i) f∗x = 0 or (ii) f∗x maps the horizontal space H = (ker f∗)⊥ conformallyonto Tf (x)N, i.e., f∗x is surjective and f∗ satisfies the equation (1.1) for U, V vectorstangent to Hx. If a point x is of type (i), then it is called critical point of f.A point x of type(ii) is called regular. The number ∧(x) is called the square dilation which is necessarilynon-negative. Its square root λ(x) = √∧(x) is known as the dilation. A horizontallyweakly conformal map f to be horizontally homothetic if the gradient of their dilation λ
is vertical, i.e., H(gradλ) = 0 at regular points. If a horizontally weakly conformal mapf has no critical points, then f is called horizontally conformal submersion [2]. Thus,it follows that a Riemannian submersion is a horizontally conformal submersion withdilation identically one. The horizontal conformal maps were introduced independentlyby Fuglede in 1978 [9] and by Ishihara 1979 [15]. From the above argument, one candetermine that the notion of horizontal conformal maps is a generalization of the conceptof Riemannian submersions.
Next, we memorize the following description in [10]. Let f : (M, gM) → (N, gN)
be a submersion. A vector field X on M is said to be projectable if there exists a vectorfield X on N , such that f∗(Xx) = Xf(x)
for each x ∈ M . In this situation X and X arecalled f −related. We call a horizontal vector field Y on (M, gM) a basic vector fieldsif it is projectable. We know that if Y is a vector field on N, then there exists a uniquebasic vector field Y on M , such that Y and Y are f −related. The vector field Y is calledthe horizontal lift of Y .
Conformal anti-invariant Submersions 3579
We denote the kernel space of f∗ by ker f∗ and consider the orthogonal complemen-tary space H = (ker f∗)⊥ to ker f∗. Then the tangent bundle of M has the followingdecomposition
T M = (ker f∗) ⊕ (ker f∗)⊥. (1.2)
We also denote the range of f∗ by rangef∗ and consider the orthogonal comple-mentary space (rangef∗)⊥ to rangef∗ in the tangent bundle T N of N. Thus the tangentbundle T N of N has the following decomposition
T N = (rangef∗) ⊕ (rangef∗)⊥. (1.3)
We know that Riemannian submersions are very special maps comparing with con-formal submersions. Although conformal maps do not preserve distance between pointscontrary to isometries, they preserve angles between vector fields. This property enablesone to transfer certain properties of a manifold to another manifold by deforming suchproperties. The concept of Conformal anti-invariant submersions from almost Hermitianmanifolds onto Riemannian manifold has been studied in [1].
In this paper, we study conformal anti-invariant submersions from Sasakian mani-folds onto Riemannian manifolds. The paper is organized as follows. In section 2, wecollect main notions and formulae which we need in this paper.
In section 3, we introduce conformal anti-invariant submersions from almost contactmetric manifolds onto Riemannian manifolds, investigate the geometry of leaves of thehorizontal distribution and the vertical distribution. In section 4, we study the necessaryand sufficient conditions for a conformal anti-invariant submersion to be harmonic andtotally geodesic. In section 5, we prove that there are certain product structures onthe total space of a conformal anti-invariant submersion from Sasakian manifold onRiemannian manifold such that ξ is vertical vector field. Finally in section 6, we givesome examples of conformal anti-invariant submersion such that the characteristic vectorfield ξ is vertical.
2. Preliminaries
An n−dimensional Riemannian manifold M is said to be an almost contact metric man-ifold, if there exist on M, a (1, 1) tensor field φ, a vector field ξ, a 1−form η andRiemannian metric g such that
φ2 = −I + η ⊗ ξ, φξ = 0, η ◦ φ = 0, (2.1)
g(X, ξ) = η(X), (2.2)
η(ξ) = 1, (2.3)
and
g(φX, φY ) = g(X, Y ) − η(X)η(Y ), g(φX, Y ) = −g(X, φY ), (2.4)
for any vector fields X, Y on M .
3580 Sushil Kumar and Rajendra Prasad
An almost contact metric manifold is said to be a contact metric manifold if dη = �,
where �(X, Y ) = g(X, φY ) is called the fundamental 2−form of M. On the otherhand the almost contact metric structure of M is said to be normal if [φ, φ](X, Y ) =−2dη(X, Y )ξ, for any X, Y, where [φ, φ] denotes the Nijenhuis tensor of φ given by
[φ, φ] (X, Y ) = φ2[X, Y ] + [φX, φY ] − φ[φX, Y ] − φ[X, φY ]. (2.5)
A normal contact metric manifold is called a Sasakian manifold [4]. It can be provedthat a Sasakian manifold is K−contact, and that an almost contact metric manifold isSasakian if and only if
(∇Xφ)Y = g(X, Y )ξ − η(Y )X, (2.6)
for any vector fields X, Y on M.
Moreover, for a Sasakian manifold the following equations satisfy:
R(X, Y )ξ = η(Y )X − η(X)Y, (2.7)
R(ξ, X)Y = g(X, Y )ξ − η(Y )X, (2.8)
∇Xξ = −φX. (2.9)
Definition 2.1. ([2]) Let (M, gM) and (N, gN) are two Riemannian manifold withRiemannian metrics gM and gN, respectively. If f : (M, gM) → (N, gN) be a differen-tiable map between given Riemannian manifolds, then f is called horizontally weaklyconformal or semi-conformal at q if either
(i) dfq = 0, or
(ii) dfq maps the horizontal space Hq = (ker(dfq))⊥ conformally onto Tf (q)N i.e.,
dfq is surjective and there exists a number �(q) = 0 such that
gN(f∗U, f∗V ) = �(q)gM(U, V ) (U, V ∈ Hq), (2.10)
where q ∈ M.
Watson introduced the fundamental tensors of a submersion in [19]. It is well knownthat the fundamental tensor play parallel role to that of the second fundamental form ofan immersion. More exactly, O’Neill defined tensors A and T for vector fields E and F
on M byAEF = V∇HEHF + H∇HEVF, (2.11)
TEF = H∇VEVF + V∇VEHF, (2.12)
where V and H are the vertical and horizontal projections [8], and ∇ is Riemannianconnection on M . On the other hand, from equations (2.11) and (2.12), we have
∇XY = TXY + ∇XY, (2.13)
Conformal anti-invariant Submersions 3581
∇XU = H∇XU + TXU, (2.14)
∇UX = AUX + V∇UX, (2.15)
∇UV = H∇UV + AUV, (2.16)
for any X, Y ∈ �(ker f∗) and U, V ∈ �(ker f∗)⊥, where V∇XY = ∇XY. If U is basic,then AXU = H∇XU.
It is Simply seen that for q ∈ M, X ∈ Vq and U ∈ Hq the linear opretors
AU, TX : TqM → TqM,
are skew-symmetric, that is
gM(AUE, F ) = −gM(E, AUF) and gM(TXE, F ) = −gM(E, TXF), (2.17)
for each E, F ∈ TqM.We have also defined the restriction of T to the vertical distributionT |V×V is precisely the second fundamental form of the fibres of f . Since TV is skew-symmetric we get: f has totally geodesic fibres if and only if T ≡ 0. For the specialevent when f is horizontally conformal we have the following proposition.
Proposition 2.2. ([11] (2.1.2)) Let f be horizontal conformal submersion betweenRiemannian manifolds (M, gM) and (N, gN) with dilation λ and U, V be horizontalvector fields, then
AUV = 1
2
{V[U, V ] − λ2gM(U, V )gradV
(1
λ2
)}, (2.18)
We know that the skew-symmetric part of A|H×H measures the obstruction integra-bility of the horizontal distribution H.
We also memorize the concept of harmonic maps between Riemannian manifolds.Let f : (M, gM) → (N, gN) is a smooth map between Riemannian manifolds. Then thedifferential f∗ of f can be observed a section of the bundle Hom(T M, f −1T N) → M,
where f −1T N is the bundle which has fibres (f −1T N)x = Tf (x)N has a connection∇ induced from the Riemannian connection ∇M and the pullback connection. Then thesecond fundamental form of f is given by
(∇f∗)(U, V ) = ∇f
Uf∗(V ) − f∗(∇MU V ), (2.19)
for vector fields U, V ∈ �(T M), where ∇f is the pullback connection. We know thatthe second fundamental form is symmetric. A smooth map f between Riemannianmanifolds is said to be harmonic if trace(∇f∗) = 0. On the extra need, the tension fieldof f is the section τ(f ) of �(f −1T N) defined by
τ(f ) = divf∗ =m∑
i=1
(∇f∗)(ei, ei), (2.20)
3582 Sushil Kumar and Rajendra Prasad
where {ei, . . . , em} is orthonormal frame on M . Then it follows that f is harmonic ifand only if τ(f ) = 0, for facts [2].
Lastly, we recollect the subsequent lemma from [2].Lemma 2.3. Let (M, gM) and (N, gN) are two Riemannian manifolds. Iff : (M, gM) →(N, gN) horizontally conformal submersion between Riemannian manifolds, then forany horizontal vector fields U, V and vertical vector fields X, Y we have
(i) ∇df (U, V ) = U(lnλ)df (V ) + V (lnλ)df (U) − gM(U, V )df (gradlnλ);(ii) ∇df (X, Y ) = −df (AV
XY);(iii) ∇df (U, X) = −df (∇M
U X) = df ((AH)∗UX).
where (AH)∗X is the adjoint of (AHX
) characterized by
〈(AH)∗UE, F 〉 = 〈E, AHU
F 〉, ( for E, F ∈ �(T M)).
3. Conformal anti-invariant submersions admitting verticalstructure vector field
In this section, we define conformal anti-invariant submersions from an almost contactmetric manifold onto Riemannian manifolds.
Definition 3.1. Let (M, φ, ξ, η, gM) be a almost contact metric manifold and (N, gN)
be a Riemannian manifold, where dimM = m and dimN = n. A horizontally confor-mal submersion f : (M, φ, ξ, η, gM) → (N, gN) is called a conformal anti-invariantsubmersion if the distribution ker f∗ is anti-invariant with respect to φ i.e., φ(ker f∗) ⊆(ker f∗)⊥. We have φ(ker f∗)⊥ ∩ ker f∗ = {0}. We denote the complementary orthonor-mal distribution to φ(ker f∗) in (ker f∗)⊥ by µ. Then we have
(ker f∗)⊥ = φ(ker f∗) ⊕ µ.
It is clear that µ is an invariant distribution of (ker f∗)⊥ under the endomorphism φ.
Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Riemannian manifold.If a map f : (M, φ, ξ, η, gM) → (N, gN) horizontally conformal submersion admittingvertical structure vector field i.e., (ξ ∈ ker f∗).
Then we have(ker f∗)⊥ = φ(ker f∗) ⊕ µ. (3.1)
It is clear that µ is an invariant distribution of (ker f∗)⊥, under the endomorphism φ.
Thus, for any U ∈ �(ker f∗)⊥, we have
φU = BU + CU, (3.2)
Conformal anti-invariant Submersions 3583
whereBU ∈ �(ker f∗) andCU ∈ �(µ).On the additional point, sincef∗(�(ker f∗)⊥) =T N and f is a conformal submersion, for every X ∈ �(ker f∗) and U ∈ (�(ker f∗)⊥),
using equation (3.2) we get1
λ2 gN(f∗φU, f∗CX) = 0, which denotes that
T N = f∗(φ(ker f∗)) ⊕ f∗(µ). (3.3)
For any vector field X ∈ �(ker f∗) and V ∈ �(ker f∗)⊥, using equations (2.1),
(2.2), (3.1) and (3.2), we get
C2V = −V − φBV, BCV = 0, η(BV ) = 0,
BφX = −X + η(X)ξ, CφX = 0, C3V + CV = 0.
Lemma 3.2. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Riemannianmanifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariant submersion,then we get
TXξ = −φX, AV ξ = −CV,
gM(CV, φX) = 0, (3.4)
andgM(∇M
U CV, φX) = −gM(CV, φAUX) + η(X)gM(U, CV ), (3.5)
for X ∈ �(ker f∗) and U, V, φX ∈ (�(ker f∗)⊥).
Proof. For X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥), since BV ∈ �(ker f∗) and φX ∈(�(ker f∗)⊥), using equations (3.2) and (2.4), we have
gM(CV, φX) = 0.
Now, using equations (2.6), (2.14) and (3.4), we get
gM(∇UCV, φX) = −gM(CV, ∇UφX)
= −gM(CV, φAUX) + η(X)gM(U, CV ),
since φV∇UX ∈ �(φ(ker f∗)). Therefore, we obtain the result. �
Theorem 3.3. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then the followings are equivalent to each other:
(i) (ker f∗)⊥is integrable,
(ii)1
λ2 gN(∇f
V f∗CU − ∇f
Uf∗CV, f∗φX)
= gM(AUBV − AV BU, φX) − gM(Hgradlnλ, CV )gM(U, φX)
+gM(Hgradlnλ, CU)gM(V, φX) − 2gM(Hgradlnλ, φX)gM(CU, V ),
3584 Sushil Kumar and Rajendra Prasad
for X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥).
Proof. For X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥), since φV ∈ (� ker f∗ ⊕ µ) andφX ∈ (�(ker f∗)⊥). Using equations (2.1), (2.4), (2.6) and (3.2) , we get
gM([U, V ], X) = gM(φ∇UV, φX) + η(X)η(∇UV ) − gM(φ∇V U, φX) − η(X)η(∇V U),
Since f is a conformal submersion, using equations (2.14) and (2.15) we get
gM([U, V ], X) = gM(AUBV − AV BU, φX) + 1
λ2 gN(f∗∇UCV, f∗φX)
− 1
λ2 gN(f∗∇V CU, f∗φX) + gM([U, V ], ξ)η(X).
Using equations (2.23), (3.4) and lemma 1(i), we get
gM([U, V ], X) = gM(AUBV − AV BU, φX) − gM(Hgradlnλ, CV )gM(U, φX)
+ gM(Hgradlnλ, CU)gM(V, φX)
− 2gM(Hgradlnλ, φX)gM(CU, V ) − 1
λ2 gN(∇f
V f∗CU
− ∇f
Uf∗CV, f∗φX) + gM([U, V ], ξ)η(X).
which implies (i) ⇔ (ii). �
Theorem 3.4. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rieman-nian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariant sub-mersion, then any two of the following conditions imply the third:
(i) (ker f∗)⊥ is integrable,
(ii) f is horizontally homothetic,
(iii)1
λ2 gN(∇f
V f∗CU − ∇f
Uf∗CV, f∗φX) = gM(AUBV − AV BU, φX),
for X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥).
Proof. For X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥), using theorem (2), we get
gM([U, V ], X) = gM(AUBV − AV BU, φX) − gM(U, φX)gM(Hgradlnλ, CV )
+ gM(V, φX)gM(Hgradlnλ, CU)
− 2gM(CU, V )gM(Hgradlnλ, φX)
− 1
λ2 gN(∇f
V f∗CU − ∇f
Uf∗CV, f∗φX) + gM([U, V ], ξ)η(X).
Conformal anti-invariant Submersions 3585
Now, using conditions (i) and (ii), we get (iii)
1
λ2 gN(∇f
V f∗CU − ∇f
Uf∗CV, f∗φX) = gM(AUBV − AV BU, φX).
Similarly, one can obtain the other assertions. �
Remark 3.5. Let f be a conformal anti-invariant submersion is conformal Lagrangiansubmersion, if φ(ker f∗) = (ker f∗)⊥. Then (3.3), we have T N = f∗(φ(ker f∗)⊥).
Corollary 3.6. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then the following assertions are equivalent to each other:
(i) (ker f∗)⊥ is integrable,
(ii) AUφV = AV φU,
(iii) (∇f∗)(V , φU) = (∇f∗)(U, φV ), for U, V ∈ (�(ker f∗)⊥).
Proof. For any X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥), from definition (2), φX ∈(�(ker f∗)⊥) and φV ∈ �(φ(ker f∗)). From theorem (2), we have
gM([U, V ], X) = gM(AUBV − AV BU, φX) − gM(Hgradlnλ, CV )gM(U, φX)
+ gM(Hgradlnλ, CU)gM(V, φX)
− 2gM(Hgradlnλ, φX)gM(CU, V )
− 1
λ2 gN(∇f
V f∗CU − ∇f
Uf∗CV, f∗φX) + η(X)gM([U, V ], ξ).
Since f conformal Lagrangian submersion, we have
gM([U, V ], X) = gM(AUBV − AV BU, φX) + η(X)gM([U, V ], ξ),
which implies (i) ⇔ (ii). On the other hand using definition (2) and equation (2.15),we get
gM(AUBV − AV BU, φX)
= gM(AUBV, φX) − gM(AV BU, φX),
= 1
λ2 gN(f∗AUBV, f∗φX) − 1
λ2 gN(f∗AV BX, f∗φX),
= 1
λ2 gN(f∗(∇UBV ), f∗φX) − 1
λ2 gN(f∗(∇V BU), f∗φX).
3586 Sushil Kumar and Rajendra Prasad
Now, using equation (2.23) we have
1
λ2 gN(f∗(∇UBV ), f∗φX) − 1
λ2 gN(f∗(∇V BU), f∗φX)
= 1
λ2 gN(−(∇f∗)(U, BV ) + ∇f
Uf∗BV, f∗φX)
− 1
λ2 gN(−(∇f∗)(V , BU) + ∇f
V f∗BU, f∗φX),
= 1
λ2 [gN((∇f∗)(V , BU) − (∇f∗)(U, BV ), f∗φX)],
which proves that (ii) ⇔ (iii). �
Theorem 3.7. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then the followings are equivalent to each other:
(i) (ker f∗)⊥ defines a totally geodesic foliation on M ,
Theorem 3.8. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then any two of the following conditions imply the third:
(i) (ker f∗)⊥ defines a totally geodesic foliation on M ,
Conformal anti-invariant Submersions 3587
(ii) f is horizontally homothetic,
(iii) gM(AUBV, φX) = − 1
λ2 gN(∇f
Uf∗CV, f∗φX), for any X ∈ �(ker f∗) and
U, V ∈ (�(ker f∗)⊥).
Proof. For X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥), using theorem (4), we have
Corollary 3.9. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal Lagrangiansubmersion, then the followings are equivalent to each other:
(i) (ker f∗)⊥ defines a totally geodesic foliation on M,
(ii) AUφV = 0,
(iii) (∇f∗)(U, φV ) = 0, for U, V ∈ (�(ker f∗)⊥).
Proof. For X ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥), from definition (2),
φV ∈ �(φ(ker f∗)) and φX ∈ �((ker f∗)⊥). Using theorem (4), we have
Since f is conformal Lagrangian submersion, we get
gM(∇UV, X) = gM(AUBV, φX) + η(X)η(∇UV ),
= gM(AUφV, φX) + η(X)η(∇UV ),
which implies (i) ⇔ (ii).
Further, using equation (2.15), we get
gM(AUBV, φX) = gM(∇UBV, φX).
3588 Sushil Kumar and Rajendra Prasad
Since f is conformal submersion, we get
gM(AUBV, φX) = 1
λ2 gN(f∗∇UBV, f∗φX).
Using equation (2.23), we get
gM(AUBV, φX) = − 1
λ2 gN((∇f∗)(U, BV ), f∗φX),
= − 1
λ2 gN((∇f∗)(U, φV ), f∗φX).
which shows (ii) ⇔ (iii). �
Theorem 3.10. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then the followings are equivalent to each other:
(i) (ker f∗) defines a totally geodesic foliation on M,
Theorem 3.11. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then any two of the following conditions imply the third:
(i) (ker f∗) defines a totally geodesic foliation on M,
(ii) λ is constant on �(µ),
(iii)1
λ2 gN(∇f
φY f∗φX, f∗φCU)
= −gM(TXφY, φU) + η(Y )gM(X, φU),
for X, Y ∈ �(ker f∗) and U ∈ (�(ker f∗)⊥).
Proof. For X, Y ∈ �(ker f∗) and U ∈ (�(ker f∗)⊥), from theorem (6), we have
From above equation λ is constant on �(µ). Similarly, one can obtain the other assertions.�
Corollary 3.12. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal Lagrangiansubmersion, then the following statements are equivalent to each other:
(i) (ker f∗) defines a totally geodesic foliation on M,
(ii) TXφY = −η(Y )X, or TXφY = −η(Y )X is parallel to ξ for X, Y ∈ �(ker f∗).
Proof. For X, Y ∈ �(ker f∗) and U ∈ �(ker f∗)⊥, from theorem (6), we have
Since f is a conformal Lagrangian submersion, we get
gM(∇XY, U) = gM(TXφY, φU) + η(Y )gM(X, φU),
which proves (i) ⇔ (ii). �
3590 Sushil Kumar and Rajendra Prasad
4. Harmonicity of conformal anti-invariant submersionadmitting vertical structure vector field
In this section, we investigate the necessary and sufficient conditions for a conformalanti-invariant submersions to be harmonic. We also find the necessary and sufficientconditions for such submersions to be totally geodesic.
Theorem 4.1. Let (M2k+2r+1, φ, ξ, η, gM) be a Sasakian manifold and (Nk+2r , gN) bea Riemannian manifold. If (M2k+2r+1, φ, ξ, η, gM) → (Nk+2r , gN) be a conformalanti-invariant submersion, then the tension field τ of f is
where µker f∗ is the mean curvature vector field of the distribution of ker f∗.
Proof. Let {e1, e2, . . . , ek, ξ , φe1, . . . , φek, µ1, . . . , µr, φµ1, . . . , φµr} be an orthonor-mal basis of �(T M) such that {e1, e2, . . . , ek, ξ} is orthonormal basis of �(ker f∗),{φe1, . . . , φek} is orthonormal basis of �(φ ker f∗) and {µ1, . . . , µr, µr+1, . . . , µ2r} isorthonormal basis of �(µ).
Then the trace of fundamental form (restriction of ker f∗ × ker f∗ ) is given by
trace(ker f∗)⊥(∇f∗) =k∑
i=1
(∇f∗)(φei, φei) +2r∑
j=1
(∇f∗)(µj , µj ).
Using lemma 1(i), we obtain
trace(ker f∗)⊥(∇f∗) =k∑
i=1
2gM(gradlnλ, φei)f∗(φei) − kf∗(gradlnλ)
+2r∑
i=1
2gM(gradlnλ, µj )f∗(µj ) − 2rf∗(gradlnλ).
Since f is conformal anti-invariant submersion, for x ∈ M, and 1 ≤ i ≤ k, 1 ≤ h ≤ r
{ 1
λ(x)f∗x(φei),
1
λ(x)f∗x(µh)} is an orthonormal basis of Tf (x)N; thus we obtain
trace(ker f∗)⊥(∇f∗) =k∑
i=1
2gN(f∗(gradlnλ),1
λf∗(φei))
1
λf∗(φei) − kf∗(gradlnλ)
+2r∑
i=1
2gN(f∗(gradlnλ),1
λf∗(µj ))
1
λf∗(µj ) − 2rf∗(gradlnλ)
trace(ker f∗)⊥(∇f∗) = (2 − k − 2r)f∗(Hgradlnλ). (4.2)
Conformal anti-invariant Submersions 3591
In a similarly, we get
trace(ker f∗)(∇f∗) =k∑
i=1
(∇f∗)(ei, ei) + (∇f∗)(ξ , ξ).
Using equation (2.22) and (2.13), we find
trace(ker f∗)(∇f∗) = −kf∗(µker f∗). (4.3)
From equations (4.2) and (4.3), we get
τ(f ) = −kf∗(µker f∗) + (2 − k − 2r)f∗(Hgradlnλ).
Therefore, we obtain the result. �
Theorem 4.2. Let (M2k+2r+1, φ, ξ, η, gM) be a Sasakian manifold and (Nk+2r , gN) bea Riemannian manifold. If (M2k+2r+1, φ, ξ, η, gM) → (Nk+2r , gN) be a conformalanti-invariant submersion, then any two of the following conditions imply the third:
(i) f is harmonic,
(ii) The fibres are minimal,
(iii) f is a horizontally homothetic map.
Proof. Taking equation (4.1), we have
τ(f ) = −kf∗(µker f∗) + (2 − k − 2r)f∗(Hgradlnλ).
Now, using conditions (i) and (ii), then f is a horizontally homothetic map. �
Corollary 4.3. Let (M2k+2r+1, φ, ξ, η, gM) be a Sasakian manifold and (Nk+2r , gN) bea Riemannian manifold. Let (M2k+2r+1, φ, ξ, η, gM) → (Nk+2r , gN) be a conformalanti-invariant submersion. If k + 2r = 2, then f is harmonic if and only if the fibres areminimal.
Next, we find necessary and sufficient condition for conformal anti-invariant sub-mersion to be totally geodesic. We memorize that a differentiable map f between twoRiemannian manifolds is called totally geodesic if
(∇f∗)(V , W) = 0, for all V, W ∈ �(T M).
A geometric clarification of a totally geodesic map is that it maps every geodesic inthe total space into a geodesic in the base space in proportion to arc lengths.
3592 Sushil Kumar and Rajendra Prasad
Theorem 4.4. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then f is a totally geodesic map if and only if
for any V, W ∈ �(T M), where W = W1+W2,W1 ∈ �(ker f∗) and W2 ∈ (�(ker f∗)⊥).
Proof. Taking equation (2.22) and using equations (2.1) and (2.6), we get
(∇f∗)(V , W) = ∇f
V f∗W + f∗(φ∇V φW + η(W)φV − η(∇V W)ξ),
for any V, W ∈ �(T M).
Now, using equations (2.15) and (3.2), we get
(∇f∗)(V , W) = ∇f
V f∗W + f∗(φAV φW1 + BH∇V φW1 + CH∇V φW1
+BAV BW2 + CAV BW2 + φV∇V BW2 + φAV CW2
+BH∇V CW2 + CH∇V CW2 + η(W1)BV
+η(W1)CV − η(∇V W)ξ),
for W = W1 + W2 ∈ �(T M), where W1 ∈ �(ker f∗) and W2 ∈ (�(ker f∗)⊥).
Thus taking into account the vertical terms, we get
(∇f∗)(V , W) = ∇f
V f∗W + f∗(φ(AV φW1 + V∇V BW2 + AV CW2)
+C(H∇V φW1 + AV BW2 + H∇V CW2) + η(W1)CV ).
Thus
(∇f∗)(V , W) = 0 ⇔−∇f
V f∗W = f∗(φ(AV φW1 + V∇V BW2 + AV CW2)
+C(H∇V φW1 + AV BW2 + H∇V CW2) + η(W1)CV ).
Therefore, we obtain the result. �
Definition 4.5. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then f is called a (φ ker f∗, µ)−totally geodesic map provided
(∇f∗)(φX, U) = 0, for X ∈ �(ker f∗) and U ∈ (�(ker f∗)⊥).
Theorem 4.6. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then f is called a (φ ker f∗, µ)−totally geodesic map if and if f is hori-zontally homothetic map.
Conformal anti-invariant Submersions 3593
Proof. For X ∈ �(ker f∗) and U ∈ �(µ), using lemma 1(i), we get
From above equation shows that λ is a constant on �(φ(ker f∗)). Thus λ is a constanton �((ker f∗)⊥). �
Theorem 4.7. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rieman-nian manifold. Let f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion. Then f is totally geodesic map if and only if
(i) TXφY = −η(Y )X and H∇Y φX ∈ �(φ ker f∗),
(ii) f is horizontally homothetic map,
(iii) ∇XBU = −TXCU or parallel to ξ and TXBU + H∇XCU ∈ �(φ ker f∗), for allX, Y ∈ �(ker f∗) and U, V ∈ �(µ).
Proof. For X, Y ∈ �(ker f∗) − {ξ}, using equations (2.1), (2.6) and (2.22), we get
(∇f∗)(X, Y ) = f∗(φ(∇XφY) + η(Y )φX − η(∇XY)ξ).
Now, using equations (2.14) and (3.2), we get
(∇f∗)(X, Y ) = f∗(φTXφY + CH∇XφY + η(Y )φX).
Thus shows that TXφY + η(Y )X = 0 and H∇XφY ∈ �(φ ker f∗).On the other hund using lemma 1(i), we get
(∇f∗)(U, V ) = U(lnλ)f∗(V ) + V (lnλ)f∗(U) − gM(U, V )f∗(gradlnλ),
for U, V ∈ �(µ). It is obvious that if f is horizontally homothetic, it follows that(∇f∗)(U, V ) = 0. Conversely, if (∇f∗)(U, V ) = 0, taking V = φU in above equation,we have
5. Decomposition Theorems for a conformal anti-invariantsubmersion admitting vertical structure vector field
In this section, we obtain decomposition theorems by using the existence of conformalanti-invariant submersions. Initial, we memorise the following results from [20]. LetgB be a Riemannian metric tensor on the manifold B = M × N and assume that thecanonical foliations DM and DN intersect perpendicularly everywhere. Then gB is themetric tensor of
(i) a twisted product M ×F N if and only if DM is totally geodesic foliation and DN
is totally umbilical foliation,
(ii) a warped product M ×F N if and only if DM is totally geodesic foliation andDN is a spheric foliation, i.e., it is umbilical and its mean curvature vector field isparallel.
Conformal anti-invariant Submersions 3595
We note in this case, from [3] we have
∇XU = X(ln F)U,
for X ∈ �(T M) and U ∈ �(T N), where ∇ is the Riemannian connection onM × N,
(iii) a usual product of Riemannian manifolds if and only if DM and DN are totallygeodesic foliations.
Next, we found a decomposition theorem related to the concept of twisted productmanifold. However, we first memorise the adjoint map of a map. Let f : (M, gM) →(N, gN)be a map between Riemannian manifolds (M, gM) and (N, gN).Then the adjointmap ∗f∗ of f∗ is characterized gM(X, f∗pY ) = gN(∗f∗pX, Y ) by X ∈ TpM, Y ∈Tf (p)N and p ∈ M. Considering f h∗ at each p ∈ M as a linear transformation
→ (T(q)N, gN(q)). The linear transformation (∗f∗p)h : (rangef∗(p)) → (ker f∗)⊥(p)
defined (∗f∗(p))hY =∗ f h
∗(p)Y, where Y ∈ (ranrgef∗(p)), q = f (p), is an isomorphism
and (f h∗(p))
−1 = (∗f∗p)h =∗ f h∗(p).
Our first decomposition theorem for a conformal anti-invariant submersion comesfrom theorem (4) and theorem (6) in terms of the second fundamental forms of suchsubmersions.
Theorem 5.1. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rieman-nian manifold. Let f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion. Then f is a locally product manifold if and only if
− 1
λ2 gN(∇f
Uf∗CV, f∗φX) (5.1)
= gM(AUBV, φX) − gM(gradlnλ, CV )gM(U, φX)
+gM(gradlnλ, φX)gM(U, CV ),
and
− 1
λ2 gN(∇f
φY f∗φX, f∗φCU) (5.2)
= gM(TXφY, BU) + gM(φY, φX)gM(Hgradlnλ, φCU)
+η(Y )gM(X,BU),
for X, Y ∈ �(ker f∗) and U, V, W ∈ (�(ker f∗)⊥).
Again, from Corollary (2) and Corollary (3), we have the following theorem.
3596 Sushil Kumar and Rajendra Prasad
Theorem 5.2. Let (M, φ, ξ, η, gM) be a Sasakian manifold and (N, gN) be a Rie-mannian manifold. If f : (M, φ, ξ, η, gM) → (N, gN) be a conformal anti-invariantsubmersion, then f is a locally product manifold if and only if AUφV = 0 andTXφY = −η(Y )X, for X, Y ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥).
Theorem 5.3. Letf be a conformal anti-invariant submersion from a Sasakian manifolds(M, φ, ξ, η, gM) to a Riemannian manifold (N, gN). Then M is a locally twisted productmanifold of the form M(ker f∗) × M(ker f∗)⊥ if and only if
− 1
λ2 gN(∇f
φY f∗φX, f∗φCU) (5.3)
= gM(TXφY, BU) + gM(φY, φX)gM(Hgradlnλ, φCU)
+η(Y )gM(X, φU),
and
gM(U, V )H = −BAUBV + CV (lnλ)BU
−B(Hgradlnλ)gM(U, CV ) (5.4)
−φ∗f∗(∇f
Uf∗CV ),
for X, Y ∈ �(ker f∗) and U, V ∈ (�(ker f∗)⊥), where M(ker f∗) and M(ker f∗)⊥ areintegral manifolds of the distributions (ker f∗)⊥ and (ker f∗) and H is the mean curvaturevector field of M(ker f∗)⊥ .
Proof. For X, Y ∈ �(ker f∗) and U ∈ (�(ker f∗)⊥), using equations (2.4), (2.6), (2.14)
Note that given an Euclidean space (x1, . . . , x2m, x2m+1) with coordinates we can canon-ically choose an almost contact structure φ on R2m+1 as follows:
φ(a1∂
∂x1+ a2
∂
∂x2+ · · · + a2m−1
∂
∂x2m−1+ a2m
∂
∂x2m
+ a2m+1∂
∂x2m+1)
= (−a2∂
∂x1+ a1
∂
∂x2+ · · · − a2m
∂
∂x2m−1+ a2m−1
∂
∂x2m
)
where ξ = ∂
∂x2m+1and a1, a2, . . . , a2m, a2m+1 are C∞−real valued functions in R. Let
η = dx2m+1 and (∂
∂x1,
∂
∂x2, . . . ,
∂
∂x2m
,∂
∂x2m+1) is orthogonal basis of vector fields on
R2m+1.
Example 6.1. Define a map f : R5 → R2 by
f (x1, . . . , x5) = (ex1 sin x2, ex1 cos x2)
Then we have
ker f∗ =<∂
∂x3,
∂
∂x4,
∂
∂x5> and (ker f∗)⊥ =<
∂
∂x1,
∂
∂x2>
Thus, f is a conformal anti-invariant submersion with λ = ex1 .
Example 6.2. Define a map f : R5 → R2 by
f (x1, . . . , x5) = (ex3 cos x4, ex3 sin x4)
Then we have
ker f∗ =<∂
∂x1,
∂
∂x2,
∂
∂x5> and (ker f∗)⊥ =<
∂
∂x3,
∂
∂x4>
Thus, f is a conformal anti-invariant submersion with λ = ex3 .
References
[1] M. A. Akyol and B. Sahin, Conformal anti-invariant submersions from almostHermitian manifolds. Turk J. Math., 40 (2016), 43–70.
[2] P. Baird and J. C. Wood, Harmonic Morphisms Between Riemannian Manifolds.London Mathematical Society Monographs, 29, Oxford University Press, TheClarendon Press. Oxford, (2003).
Conformal anti-invariant Submersions 3599
[3] RL. Bishop and B. O’Neill, Manifolds of negative curvature, Trans Amer MathSoc. 145 (1969), 1–49.
[4] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Birkhi-iuser, Boston, (2002).
[5] J. P. Bourguignon and H. B. Lawson, A Mathematician’s visit to Kaluza-Kleintheory, Rend. Semin. Mat. Torino Fasc. Spec. (1989), 143–163.
[7] D. Chinea, Almost contact metric submersions, Rend. Circ. Mat. Palermo, 34(1)(1985), 89–104.
[8] M. Falcitelli, S. Ianus and A. M. Pastore, Riemannian submersions and relatedtopics, World Scientific Publishing Co., (2004).
[9] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst.Fourier (Grenoble) 28 (1978), 107–144.
[10] S. Gudmundsson, The Geometry of Harmonic Morphisms, PhD Thesis, Universityof Leeds, Leeds, UK, (1992).
[11] D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie im Groβen,Lecture Notes in Mathematics 55, Springer (1975).
[12] S. Ianus, A. M. Ionescu, R. Mazzocco and G. E. Vilcu, Riemannian submersionsfrom almost contact metric manifolds, arXiv: 1102.1570v1 [math. DG].
[13] S. Ianus and M. Visinescu, Kaluza-Klein theory with scalar fields and generalizedHopf manifolds, Class. Quantum Gravity 4 (1987), 1317–1325.
[14] S. Ianus and M. Visinescu, Space-time compactification and Riemannian submer-sions, In: Rassias, G.(ed.), The Mathematical Heritage of C. F. Gauss, (1991),358–371, World Scientific, River Edge.
[15] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic func-tions, J. Math. Kyoto. Univ.19 (1979), 215–229.
[16] J. C. Marrero and J. Rocha, Locally conformal Kähler submersions, Geom. Dedi-cata, 52(3), (1994), 271–289.
[17] M. T. Mustafa, Applications of harmonic morphisms to gravity, J. Math. Phys.,41(10) (2000), 6918–6929.
[18] C. Murathan and I. K. Erken, Anti-invariant Riemannian submersions from Cosym-plectic manifolds, Filomat 29(7) (2015), 1429–1444.
[19] B. O’Neill, The fundamental equations of a submersion, Mich. Math. J., 13, (1966),458–469.
[20] R. Ponge and H. Reckziegel, Twisted products in pseudo-Riemannian geometry,Geom Dedicata 48 (1993), 15–25.
3600 Sushil Kumar and Rajendra Prasad
[21] B. Sahin, Anti-invariant Riemannian submersions from almost Hermitian mani-folds, Cent. Eur. J. Math. 8(3), (2010) 437–447.
[22] B. Sahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc.Sci. Math. Roumanie Tome 54(102) No 1, (2011), 93–105.
[23] B. Sahin, Semi-invariant submersions from almost Hermitian manifolds, Canad.Math. Bull., 56(1), (2013), 173–183.
[24] B. Watson, G, G|-Riemannian submersions and nonlinear gauge field equations ofgeneral relativity, In: Rassias, T. (ed.) Global Analysis - Analysis on manifolds,dedicated M. Morse. Teubner-Texte Math., 57 (1983), 324–349, Teubner, Leipzig.
[25] B. Watson, Almost Hermitian submersions, J. Differential Geometry, 11(1), (1976),147–165.
