Open access peer-reviewed chapter

# On Conformal Anti-Invariant Submersions Whose Total Manifolds Are Locally Product Riemannian

Written By

Mehmet Akif Akyol

Reviewed: July 16th, 2018 Published: April 30th, 2019

DOI: 10.5772/intechopen.80337

From the Edited Volume

## Manifolds II

Edited by Paul Bracken

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## Abstract

The aim of this chapter is to study conformal anti-invariant submersions from almost product Riemannian manifolds onto Riemannian manifolds as a generalization of anti-invariant Riemannian submersion which was introduced by B. Sahin. We investigate the integrability of the distributions which arise from the definition of the new submersions and the geometry of foliations. Moreover, we find necessary and sufficient conditions for this submersion to be totally geodesic and in order to guarantee the new submersion, we mention some examples of such submersions.

### Keywords

• conformal submersion
• almost product Riemannian manifold
• vertical distribution
• conformal anti-invariant submersion
• 2010 Mathematics Subject Classification: primary 53C15; secondary 53C40

## 1. Introduction

Immersions and submersions, which are special tools in differential geometry, also play a fundamental role in Riemannian geometry, especially when the involved manifolds carry an additional structure (such as contact, Hermitian and product structure). In particular, Riemannian submersions (which we always assume to have connected fibers) are fundamentally important in several areas of Riemannian geometry. For instance, it is a classical and important problem in Riemannian geometry to construct Riemannian manifolds with positive or non-negative sectional curvature. Riemannian submersions between Riemannian manifolds are important geometric structures. Riemannian submersions between Riemannian manifolds were studied by O’Neill [1] and Gray [2]. In [3], the Riemannian submersions were considered between almost Hermitian manifolds by Watson under the name of almost Hermitian submersions. In this case, the Riemannian submersion is also an almost complex mapping and consequently the vertical and horizontal distributions are invariant with respect to the almost complex structure of the total manifold of the submersion. The study of anti-invariant Riemannian submersions from almost Hermitian manifolds was initiated by Şahin [4]. In this case, the fibers are anti-invariant with respect to the almost complex structure of the total manifold. This notion extended to different total spaces see: [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

On the other hand, as a generalization of Riemannian submersion, horizontally conformal submersions are defined as follows [15]: Suppose that MgM and BgB are Riemannian manifolds and π:MB is a smooth submersion, then π is called a horizontally conformal submersion, if there is a positive function λ such that

λ2gMXY=gBπXπY

for every X,YΓkerπ. It is obvious that every Riemannian submersion is a particular horizontally conformal submersion with λ=1. We note that horizontally conformal submersions are special horizontally conformal maps which were introduced independently by Fuglede [16] and Ishihara [17]. We also note that a horizontally conformal submersion π:MB is said to be horizontally homothetic if the gradient of its dilation λ is vertical, i.e.,

at pM, where H is the projection on the horizontal space kerπ. For conformal submersion, see: [15, 18, 19].

One can see that Riemannian submersions are very special maps comparing with conformal submersions. Although conformal maps do not preserve distance between points contrary to isometries, they preserve angles between vector fields. This property enables one to transfer certain properties of a manifold to another manifold by deforming such properties.

Recently, we introduced conformal anti-invariant submersions [20] and conformal semi-invariant submersions [21] from almost Hermitian manifolds, and gave examples and investigated the geometry of such submersions (see also [22, 23]). We showed that the geometry of such submersions is different from their counterpart anti-invariant Riemannian submersions and semi-invariant Riemannian submersions. In the present paper, we define and study conformal anti-invariant submersions from almost product Riemannian manifolds, give examples and investigate the geometry of the total space and the base space for the existence of such submersions.

Our work is structured as follows: Section 2 is focused on basic facts for conformal submersions and almost product Riemannian manifolds. The third section is concerned with definition of conformal anti-invariant submersions, investigating the integrability conditions of the horizontal distribution and the vertical distribution. In Section 4, we study the geometry of leaves of the horizontal distribution and the vertical distribution. In Section 5, we find necessary and sufficient conditions for a conformal anti-invariant submersion to be totally geodesicness. The last section, we give some examples of such submersions.

## 2. Preliminaries

In this section we recall several notions and results which will be needed throughout the chapter.

Let M be a m-dimensional manifold with a tensor F of a type (1,1) such that

F2=I,FI.

Then, we say that M is an almost product manifold with almost product structure F. We put

P=12I+F,Q=12IF.

Then we get

P+Q=I,P2=P,Q2=Q,PQ=QP=0,F=PQ.

Thus P and Q define two complementary distributions P and Q. We easily see that the eigenvalues of F are +1 or −1. If an almost product manifold M admits a Riemannian metric g such that

gFXFY=gXYE2

for any vector fields X and Y on M, then M is called an almost product Riemannian manifold, denoted by MgF. Denote the Levi-Civita connection on M with respect to g by . Then, M is called a locally product Riemannian manifold [24] if F is parallel with respect to , i.e.,

XF=0,XΓTM.E3

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 ([15]) Let φ:MmgNnh be a smooth map between Riemannian manifolds, and let xM. Then φ is called horizontally weakly conformal or semi conformal at x if either

1. dφx=0, or

2. dφx maps horizontal space Hx=kerdφx conformally onto TφN, i.e., dφx is surjective and there exists a number Λx0 such that

hdφxXdφxY=ΛxgXYXYHx.E4

Note that we can write the last equation more sufficiently as

φhxHx×Hx=ΛxgxHx×Hx.

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, dφx has rank 0; at a regular point, dφx has rank n and φ is submersion. The number Λx is called the square dilation (of φ at x); it is necessarily non-negative; its square root λx=Λx is called the dilation (of φ at x). The map φ is called horizontally 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 [18]. Let π:MN be a submersion. A vector field E on M is said to be projectable if there exists a vector field E on N, such that Ex=Eπx for all xM. In this case E and E are called π related. A horizontal vector field Y on Mg is called basic, if it is projectable. It is well known fact, that is, Z is a vector field on N, then there exists a unique basic vector field Z on M, such that Z and Z are π related. The vector field Z is called the horizontal lift of Z.

The fundamental tensors of a submersion were introduced in [1]. 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,G on M by

AEG=VHEM1HG+HHEM1VGE5
TEG=HVEM1VG+VVEM1HGE6

where V and H are the vertical and horizontal projections (see [25]). On the other hand, from (5) and (6), we have

VM1W=TVW+̂VWE7
VM1X=HVM1X+TVXE8
XM1V=AXV+VXM1VE9
XM1Y=HXM1Y+AXYE10

for X,YΓkerπ and V,WΓkerπ, where ̂VW=VVM1W. If X is basic, then HVM1X=AXV. It is easily seen that for xM, XHx and Vx the linear operators TV, AX:TXMTXM are skew-symmetric, that is

gTVEG=gETVGandgAXEG=gEAXG

for all E,GTxM. We also see that the restriction of T to the vertical distribution TV×V is exactly the second fundamental form of the fibers of π. Since TV is skew symmetric, we get π which has totally geodesic fibers if and only if T0. For the special case when π is horizontally conformal we have the following:

Proposition 2.1 ([18]) Let π:MmgNnh be a horizontally conformal submersion with dilation and X,Y be horizontal vectors, then

We see that the skew-symmetric part of Akerπ×kerπ measures the obstruction integrability of the horizontal distribution kerπ.

Let MgM and NgN be Riemannian manifolds and suppose that π:MN is a smooth map between them. The differential of π of π can be viewed a section of the bundle HomTMπ1TNM, where π1TN is the pullback bundle which has fibers π1TNp=TπpN, pM. HomTMπ1TN has a connection induced from the Levi-Civita connection M and the pullback connection. Then the second fundamental form of π is given by

π:ΓTM×ΓTMΓTN

defined by

πXY=XππYπXMYE12

for X,YΓTM, where π is the pullback connection. It is known that the second fundamental form is symmetric.

Lemma 2.1. [26] Let MgM and NgN be Riemannian manifolds and suppose that φ:MN is a smooth map between them. Then we have

XφφYYφφXφXY=0E13

for X,YΓTM.

Finally, we recall the following lemma from [15].

Lemma 2.2. Suppose that π:MN is a horizontally conformal submersion. Then, for any horizontal vector fields X,Y and vertical fields V,W we have.

2. πVW=πTVW;

3. πXV=πXMV=πAXV.

## 3. Conformal anti-invariant submersions from almost product Riemannian manifolds

In this section, we define conformal anti-invariant submersions from an almost product Riemannian manifold onto a Riemannian manifold, investigating the geometry of distributions kerπ and kerπ and obtain the integrability conditions for the distribution kerπ for such submersions.

Definition 3.1. Let M1g1F be an almost product Riemannian manifold and M2g2 be a Riemannian manifold. A horizontally conformal submersion π:M1M2 with dilation λ is called conformal anti-invariant submersion if the distribution kerπ is anti-invariant with respect to F, i.e., Fkerπkerπ.

Let π:M1g1FM2g2 is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1F to a Riemannian manifold M2g2. First of all, from Definition 3.1, we have Fkerπkerπ0. We denote the complementary orthogonal distribution to Fkerπ in kerπ by μ. Then we have

kerπ=Fkerπμ.E14

Proposition 3.1. Let M1g1F be an almost product Riemannian manifold and M2g2 be a Riemannian manifold. Then μ is invariant with respect to F.

Proof. For ZΓμ and VΓkerπ, by using (2), we have g1FZFV=0, which show that FZ is orthogonal to Fkerπ. On the other hand, since FV and Z are orthogonal we get g1FVZ=g1VFZ=0 which shows that FZ is orthogonal to kerπ. This completes proof.□

For ZΓkerπ, we have

FZ=BZ+CZ,E15

where BZΓkerπ and CZΓμ. On the other hand, since πkerπ=TM2 and π is a conformal submersion, using (15) we derive 1λ2g2πFVπCZ=0 for any ZΓkerπ and VΓkerπ, which implies that

TM2=πFkerππμ.E16

Lemma 3.1. Let π be a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. Then we have

g1CWFV=0E17

and

g1ZM1CWFV=g1CWFAZVE18

for Z,WΓkerπ and VΓkerπ.

Proof. For WΓkerπ and VΓkerπ, using (2) we have

g1CWFV=g1FWBWFV=g1FWFV

due to BWΓkerπ and FVΓkerπ. Hence g1FWFV=g1WV=0 which is (17). Since M1 is a locally product Riemannian manifold, differentiating (3.4) with respect to Z, we get

g1ZM1CWFV=g1CWFZM1V

for Z,WΓkerπ and VΓkerπ. Then using (9) we have

g1ZM1CWFV=g1CWFAZVg1CWFVZM1V.

Since FVZM1VΓFkerπ, we obtain (18).□

We now study the integrability of the distribution kerπ and then we investigate the geometry of the leaves of kerπ and kerπ. We note that it is known that the distribution kerπ is integrable.

Theorem 3.1. Let π:M1g1FM2g2 is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1F to a Riemannian manifold M2g2. Then the following assertions are equivalent to each other;

1. kerπ is integrable,

2. 1λ2g2WππCZZππCWπFV=g1AZBWAWBZCWlnλZ+CZlnλWFVE19

for any Z,WΓkerπ and VΓkerπ.

Proof. For WΓkerπ and VΓkerπ, we see from Definition 3.1, FVΓkerπ and FWΓkerπμ. Thus using (2) and (3), for ZΓkerπ we obtain

g1ZWV=g1ZM1FWFVg1WM1FZFV.

Further, from (15) we get

g1ZWV=g1ZM1BWFV+g1ZM1CWFVg1WM1BZFVg1WM1CZFV.

Using (9), (10) and if we take into account π is a conformal submersion, we arrive at

g1ZWV=g1AZBWAWBZFV+1λ2g2πZM1CWπFV1λ2g2πWM1CZπFV.

Thus, from (12) and Lemma 2.2 we derive

Moreover, using (17), we obtain

g1ZWV=g1AZBWAWBZCWlnλZ+CZlnλWFV1λ2g2WππCZZππCWπFV

which proves ab. □

From Theorem 3.1, we deduce the following characterization.

Theorem 3.2. Let π be a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. Then any two conditions below imply the three;

1. kerπ is integrable.

2. λ is a constant on Γμ.

3. g2WππCZZππCWπFV=λ2g1AZBWAWBZFV

for Z,WΓkerπ and VΓkerπ.

Proof. From Theorem 3.1, we have

g1ZWV=g1AZBWAWBZCWlnλZ+CZlnλWFV1λ2g2WππCZZππCWπFV.

for Z,WΓkerπ and VΓkerπ. Now, if we have (i) and (iii), then we arrive at

Now, taking W=FV in (20) for VΓkerπ, using (17), we get

Hence λ is a constant on Γμ. Similarly, one can obtain the other assertions. □

We say that a conformal anti-invariant submersion is a conformal Lagrangian submersion if Fkerπ=kerπ. From Theorem 3.1, we have the following result.

Corollary 3.1. Let π be a conformal Lagrangian submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. Then the following assertions are equivalent to each other:

1. kerπ is integrable

2. AZFW=AWFZ

3. πZFW=πWFZ

for Z,WΓkerπ.

Proof. For Z,WΓkerπ and VΓkerπ, we see from Definition 3.1, FVΓkerπ and FWΓkerπ. From Theorem 3.1 we have

g1ZWV=g1AZBWAWBZCWlnλZ+CZlnλWFV1λ2g2WππCZZππCWπFV.

Since π is a conformal Lagrangian submersion, we derive

g1ZWV=g1AZBWAWBZFV

which shows iii. On the other hand, using Definition 3.1 and (9) we arrive at

g1AZBWFVg1AWBZFV=1λ2g2πAZBWπFV1λ2g2πAWBZπFV=1λ2g2πZM1BWπFV1λ2g2πWM1BZπFV.

Now, using (12) we obtain

1λ2{ g2(π(ZM1BW),πFV)g2(π(WM1BZ),πFV) }=1λ2g2((F)(Z,BW)+ZππBW,πFV)1λ2g2((F)(W,BZ)+WππBZ,πFV).

Since BZ,BWΓkerπ, we derive

g1AZBWFVg1AWBZFV=1λ2g2FWBZFZBWπFV

which tells that iiiii.

## 4. Totally geodesic foliations

In this section, we shall investigate the geometry of leaves of kerπ and kerπ. For the geometry of leaves of the horizontal distribution kerπ, we have the following theorem.

Theorem 4.1. Let π:M1g1FM2g2 is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1F to a Riemannian manifold M2g2. Then the following assertions are equivalent to each other;

1. kerπ defines a totally geodesic foliation on M1.

2. 1λ2g2ZππCWπFV=g1AZBWCWlnλZ+g1ZCWlnλFV

for Z,WΓkerπ and VΓkerπ.

Proof. For Z,WΓkerπ and VΓkerπ, by using (3), (9), (10), (14) and (15) we have

g1ZM1WV=g1AZBWFV+g1ZM1CWFV.

Since π is a conformal submersion, using (12) and Lemma 2.2 we arrive at

Moreover, using Definition 3.1 and (17) we obtain

g1ZM1WV=g1AZBWCWlnλZ+g1ZCWlnλFV+1λ2g2ZππCWπFV

which proves iii. □

From Theorem 4.1, we also deduce the following characterization.

Theorem 4.2. Let π be a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. Then any two conditions below imply the three;

1. kerπ defines a totally geodesic foliation on M1.

2. π is horizontally homothetic submersion.

3. g2ZππCWπFV=λ2g1AZFVBW

for Z,WΓkerπ and VΓkerπ.

Proof. For Z,WΓkerπ and VΓkerπ, from Theorem 4.1, we have

g1ZM1WV=g1AZBWCWlnλZ+g1ZCWlnλFV+1λ2g2ZππCWπFV.

Now, if we have (i) and (iii), then we obtain

Now, taking Z=CW) in (4.1) and using (17), we get

Thus, λ is a constant on ΓFkerπ. On the other hand, taking Z=FV in (25) and using (17) we derive

From above equation, λ is a constant on Γμ. Similarly, one can obtain the other assertions. □

For conformal Lagrangian submersion, we have the following result.

Corollary 4.1. Let π be a conformal Lagrangian submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. Then the following assertions are equivalent to each other;

1. kerπ defines a totally geodesic foliation on M1.

2. AZBW=0

3. πZFV=0

for Z,WΓkerπ and VΓkerπ.

Proof. For Z,WΓkerπ and VΓkerπ, from Theorem 4.1, we have

g1ZM1WV=g1AZBWCWlnλZ+g1ZCWlnλFV+1λ2g2ZππCWπFV.

Since π is a conformal Lagrangian submersion, we derive

g1ZM1WV=g1AZBWFV

which shows iii. On the other hand, using Definition 3.1 and (9) we arrive at

g1AZBWFV=1λ2g2πAZBWπFV=1λ2g2πZM1BWπFV.

Now, using (12) we obtain

1λ2g2πZM1BWπFV=1λ2g2πZBW+ZππBWπFV=1λ2g2πZBWπFV

which tells that iiiii.

For the totally geodesicness of the foliations of the distribution kerπ.

Theorem 4.3. Let π:M1g1FM2g2 is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1F to a Riemannian manifold M2g2. Then the following assertions are equivalent to each other;

1. kerπ defines a totally geodesic foliation on M1.

for V,UΓkerπ and ZΓkerπ.

Proof. For ZΓkerπ and V,UΓkerπ, by using (2), (3), (8) and (15) we get

g1VM1UZ=g1TVFUBZ+g1HVM1FUCZ.

Since M1 is torsion free and VFUΓkerπ we obtain

g1VM1UZ=g1TVFUBZ+g1FUM1VCZ.

Using (3) and (10) we have

g1VM1UZ=g1TVFUBZ+g1FUM1FVFCZ

here we have used that μ is invariant. Since π is a conformal submersion, using (12) and Lemma 2.2 we obtain

Moreover, using Definition 3.1 and (17), we obtain

which proves iii. □

From Theorem 4.3, we deduce the following result.

Theorem 4.4. Let π be a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. Then any two conditions below imply the three;

1. kerπ defines a totally geodesic foliation on M1

2. λ is a constant on Γμ

3. 1λ2g2FUππFVπFCZ=g1TVFUBZ

for V,UΓkerπ and ZΓkerπ.

Proof. For V,UΓkerπ and ZΓkerπ, from Theorem 4.3 we have

Now, if we have (i) and (iii), then we obtain

From above equation, λ is a constant on Γμ. Similarly, one can obtain the other assertions.□

If π is a conformal Lagrangian submersion, then (16) implies that TM2=πFkerπ. Hence we have the following corollary:

Corollary 4.2. Let π be a conformal Lagrangian submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. Then the following assertions are equivalent to each other;

1. kerπ defines a totally geodesic foliation on M1.

2. TVFU=0

for V,UΓkerπ and ZΓkerπ.

Proof. From Theorem 4.3 we have

for V,UΓkerπ and ZΓkerπ. Since π is a conformal Lagrangian submersion, we get

g1VM1UZ=g1TVFUBZ

which shows iii.

## 5. Totally geodesicness of the conformal anti-invariant submersion

In this section, we shall examine the totally geodesicness of a conformal anti-invariant submersion. We give a necessary and sufficient condition for a conformal anti-invariant submersion to be totally geodesic map. Recall that a smooth map π between two Riemannian manifolds is called totally geodesic if π=0 [15].

Theorem 5.1. Let π:M1g1FM2g2 is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1F to a Riemannian manifold M2g2. π is totally geodesic map if and only if.

1. π is a horizontally homothetic map,

2. TUFV=0 and HUM1FVΓFkerπ,

3. AZFV=0 and HZM1FVΓFkerπ

for Z,W,ZΓkerπ and U,VΓkerπ.

Proof. (a) For any Z,WΓμ, from Lemma 2.2 we derive

It is obvious that if π is a horizontally homothetic map, it follows that πZW=0. Conversely, if πZW=0, taking W=FZ in above equation, we get

Taking inner product in (31) with πFZ, we obtain

From (32), λ is a constant on Γμ. On the other hand, for U,VΓkerπ, from Lemma 2.2 we have

Again if π is a horizontally homothetic map, then πFUFV=0. Conversely, if πFUFV=0, putting U instead of V in above equation, we derive

Taking inner product in (33) with πFU and since π is a conformal submersion, we have

From above equation, λ is a constant on ΓFkerπ. Thus λ is a constant on Γkerπ.

1. (b) For any U,VΓkerπ, using (3) and (12) we have

πUV=UππVπUM1V=πFUM1FV.

Then from (7) and (8) we arrive at

πUV=πFTUFV+CHUM1FV.

From above equation, πUV=0 if and only if

πFTUFV+CHUM1FV=0E25

Since π is non-singular, this implies TUFV=0 and HUM1FVΓFkerπ.

1. (c) For ZΓμ and VΓkerπ, from (3) and (12) we get

πZV=ZππVπZM1V=πFZM1FV.

Using (9) and (10) we have

πZV=πFAZFV+CHZM1FV.

Thus πZV=0 if and only if

πFAZFV+CHZM1FV=0.

Then, since π is a linear isomorphism between kerπ and TM2, πZV=0 if and only if AZFV=0 and HZM1FVΓFkerπ. Thus proof is complete.□

Here we present another result on conformal anti-invariant submersion to be totally geodesic.

Theorem 5.2 Let π be a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1F onto a Riemannian manifold M2g2. If π is a totally geodesic map then

ZππW2=πFAZFW1+VZM1BW2+AZCW2+CHZM1FW1+AZBW2+HZM1CW2

for any ZΓkerπ and W=W1+W2ΓTM, where W1Γkerπ and W2 Γkerπ.

Proof. Using (3) and (12) we have

πZW=ZππWπFZM1FW

for any ZΓkerπ and WΓTM1. Then from (9), (10) and (15) we get

(π)(Z,W)=ZππW2π( FAZFW1+BHZM1FW1+CHZM1FW1+BAZBW2 +CAZBW2+FVZM1BW2+FAZCW2+BHZM1CW2+CHZM1CW2 )

for any W=W1+W2ΓTM, where W1Γkerπ and W2 Γkerπ. Thus taking into account the vertical parts, we find

πZW=ZππW2πFAZFW1+VZM1BW2+AZCW2+CHZM1FW1+AZBW2+HZM1CW2

which gives our assertion.□

## 6. Examples

In this section, we now give some examples for conformal anti-invariant submersions from almost product Riemannian manifolds.

Example 6.1. Every anti-invariant Riemannian submersion is a conformal anti-invariant submersion with λ=I, where I is the identity function [7].

We say that a conformal anti-invariant submersion is proper if λI. We now present an example of a proper conformal anti-invariant submersion. Note that given an Euclidean space R4 with coordinates x1x4, we can canonically choose an almost product structure F on R4 as follows:

Fa1x1+a2x2+a3x3+a4x4=a3x1+a4x2+a1x3+a2x4,
a1,,a4R.E26

Example 6.2. Let π be a submersion defined by

π:R4x1x2x3x4R2cosx1sinhx2sinx1coshx2.

Then it follows that

kerπ=spanV1=x3V2=x4

and

kerπ=spanX1=x1X2=x2.

Hence, we have FV1=X1 and FV2=X2 imply that Fkerπ=kerπ. Also by direct computations, we get

πX1=sinx1sinhx2y1+cosx1coshx2y2,πX2=cosx1coshx2y1+sinx1sinhx2y2.

Hence, we have

g2πX1πX1=sin2x1sinh2x2+cos2x1cosh2x2g1X1X1,g2πX2πX2=sin2x1sinh2x2+cos2x1cosh2x2g1X2X2,

where g1 and g2 denote the standard metrics (inner products) of R4 and R2. Thus π is a conformal anti-invariant submersion with λ2=sin2x1sinh2x2+cos2x1cosh2x2.

Example 6.3. Let π be a submersion defined by

π:R4x1x2x3x4R2ex3sinx42ex3cosx42.

Then it follows that

kerπ=spanV1=x1V2=x2

and

kerπ=spanW1=x3W2=x4.

Hence we have FV1=W1 and FV2=W2 imply that Fkerπ=kerπ. Also by direct computations, we get

πW1=ex3sinx42y1+ex3cosx42y2,πW2=ex3cosx42y1ex3sinx42y2.

Hence, we have

g2πW1πW1=ex322g1W1W1,g2πW2πW2=ex322g1W2W2,

where g1 and g2 denote the standard metrics (inner products) of R4 and R2. Thus π is a conformal anti-invariant submersion with λ=ex32.

## References

1. 1. O’Neill B. The fundamental equations of a submersion. Michigan Mathematical Journal. 1966;13:458-469
2. 2. Gray A. Pseudo-Riemannian almost product manifolds and submersions. Journal of Mathematics and Mechanics. 1967;16:715-737
3. 3. Watson B. Almost Hermitian submersions. Journal of Differential Geometry. 1976;11(1):147-165
4. 4. Şahin B. Anti-invariant Riemannian submersions from almost Hermitian manifolds. Central European Journal of Mathematics. 2010;3:437-447
5. 5. Beri A, Erken İK, Murathan C. Anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. 2016;40:540-552
6. 6. Cengizhan M, Erken IK. Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian submersions. Univerzitet u Nišu. 2015;29(7):1429-1444
7. 7. Gündüzalp Y. Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds. Journal of Function Spaces and Applications. 2013;7. Article ID 720623
8. 8. Gündüzalp Y. Anti-invariant Riemannian submersions from almost product Riemannian manifolds. Mathematical Science and Applications E-notes. 2013;1(1):58-66
9. 9. Lee JW. Anti-invariant ξ-Riemannian submersions from almost contact manifolds. Hacettepe Journal of Mathematics and Statistics. 2013;42(3):231-241
10. 10. Lee JC, Park JH, Şahin B, Song DY. Einstein conditions for the base space of anti-invariant Riemannian submersions and Clairaut submersions. Taiwanese Journal of Mathematics. In press. DOI: 10.11650/tjm.19.2015.5283
11. 11. Park KS. H-anti-invariant submersions from almost quaternionic Hermitian manifolds. arXiv:1507.04473 [math.DG]
12. 12. Shahid A, Tanveer F. Anti-invariant Riemannian submersions from nearly Kählerian mani-folds. Univerzitet u Nišu. 2013;27(7):1219-1235
13. 13. Şahin B. Riemannian submersions from almost Hermitian manifolds. Taiwanese Journal of Mathematics. 2012;17(2):629-659
14. 14. Taştan HM. On Lagrangian submersions. Hacettepe Journal of Mathematics and Statistics. 2014;43(6):993-1000
15. 15. Baird P, Wood JC. Harmonic Morphisms between Riemannian Manifolds, London Mathematical Society Monographs, 29. Oxford: Oxford University Press, The Clarendon Press; 2003
16. 16. Fuglede B. Harmonic morphisms between Riemannian manifolds. Annales De L'Institut Fourier. 1978;28:107-144
17. 17. Ishhara T. A mapping of Riemannian manifolds which preserves harmonic functions. Journal of Mathematics of Kyoto University. 1979;19:215-229
18. 18. Gundmundsson S. The geometry of harmonic morphisms [Ph.D. thesis]. University of Leeds; 1992
19. 19. Ornea L, Romani G. The fundamental equations of a conformal submersions. Beitrague Z. Algebra and Geometrie/Contributions Algebra and Geometry. 1993;34(2):233-243
20. 20. Akyol MA, Şahin B. Conformal anti-invariant submersions from almost Hermitian manifolds. Turkish Journal of Mathematics. 2016;40:43-70
21. 21. Akyol MA, Şahin B. Conformal semi-invariant submersions. Communications in Contemporary Mathematics. 2017;19(2):1650011. DOI: 10.1142/S0219199716500115
22. 22. Akyol MA. Conformal anti-invariant submersions from cosymplectic manifolds. Hacettepe Journal of Mathematics and Statistics. 2017;46(2):177-192
23. 23. Akyol MA. Conformal semi-invariant submersions from almost product Riemannian manifolds. Acta Mathematica Vietnamica. 2017;42(3):491-507. DOI: 10.1007/s40306-016-0193-9
24. 24. Yano K, Kon M. Structures on Manifolds. Singapore: World Scientific; 1984
25. 25. Falcitelli M, Ianus S, Pastore AM. Riemannian Submersions and Related Topics. River Edge, NJ: World Scientific; 2004
26. 26. Urakawa H. Calculus of variations and harmonic maps. American Mathematical Society. 1993;132

Written By

Mehmet Akif Akyol

Reviewed: July 16th, 2018 Published: April 30th, 2019