Open access peer-reviewed chapter

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

By Mehmet Akif Akyol

Submitted: April 23rd 2018Reviewed: July 16th 2018Published: April 30th 2019

DOI: 10.5772/intechopen.80337

Downloaded: 120

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 MgMand BgBare Riemannian manifolds and π:MBis a smooth submersion, then πis called a horizontally conformal submersion, if there is a positive function λsuch 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:MBis said to be horizontally homothetic if the gradient of its dilation λis vertical, i.e.,

Hgradλ=0E1

at pM, where His 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 Xand Yon M,then Mis called an almost product Riemannian manifold, denoted by MgF.Denote the Levi-Civita connection on Mwith respect to gby .Then, Mis called a locally product Riemannian manifold [24] if Fis 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:MmgNnhbe a smooth map between Riemannian manifolds, and let xM. Then φis called horizontally weakly conformal or semi conformal at xif either

  1. dφx=0, or

  2. dφxmaps horizontal space Hx=kerdφxconformally onto TφN, i.e., dφxis surjective and there exists a number Λx0such that

    hdφxXdφxY=ΛxgXYXYHx.E4

Note that we can write the last equation more sufficiently as

φhxHx×Hx=ΛxgxHx×Hx.

A point xis 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, dφxhas rank 0; at a regular point, dφxhas rank nand φis submersion. The number Λxis called the square dilation (of φat x); it is necessarily non-negative; its square root λx=Λxis called the dilation (of φat x). The map φis 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 φhas no critical points, then we call it a (horizontally) conformal submersion.

Next, we recall the following definition from [18]. Let π:MNbe a submersion. A vector field Eon Mis said to be projectable if there exists a vector field Eon N, such that Ex=Eπxfor all xM. In this case Eand Eare called πrelated. A horizontal vector field Yon Mgis called basic, if it is projectable. It is well known fact, that is, Zis a vector field on N, then there exists a unique basic vector field Zon M, such that Zand Zare πrelated. The vector field Zis 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 Tand Adefined for vector fields E,Gon Mby

AEG=VHEM1HG+HHEM1VGE5
TEG=HVEM1VG+VVEM1HGE6

where Vand Hare 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,YΓkerπand V,WΓkerπ, where ̂VW=VVM1W. If Xis basic, then HVM1X=AXV. It is easily seen that for xM, XHxand Vxthe linear operators TV, AX:TXMTXMare skew-symmetric, that is

gTVEG=gETVGandgAXEG=gEAXG

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

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

AXY=12VXYλ2gXYgradV1λ2.E11

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

Let MgMand NgNbe Riemannian manifolds and suppose that π:MNis a smooth map between them. The differential of πof πcan be viewed a section of the bundle HomTMπ1TNM, where π1TNis the pullback bundle which has fibers π1TNp=TπpN, pM. HomTMπ1TNhas a connection induced from the Levi-Civita connection Mand the pullback connection. Then the second fundamental form of πis given by

π:ΓTM×ΓTMΓTN

defined by

πXY=XππYπXMYE12

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

Lemma 2.1. [26] Let MgMand NgNbe Riemannian manifolds and suppose that φ:MNis 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:MNis a horizontally conformal submersion. Then, for any horizontal vector fields X,Yand vertical fields V,Wwe have.

  1. πXY=XlnλπY+YlnλπXgMXYπgradlnλ;

  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 kerπand kerπand obtain the integrability conditions for the distribution kerπfor such submersions.

Definition 3.1. Let M1g1Fbe an almost product Riemannian manifold and M2g2be a Riemannian manifold. A horizontally conformal submersion π:M1M2with dilation λis called conformal anti-invariant submersion if the distribution kerπis anti-invariant with respect to F,i.e., Fkerπkerπ.

Let π:M1g1FM2g2is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1Fto a Riemannian manifold M2g2.First of all, from Definition 3.1, we have Fkerπkerπ0.We denote the complementary orthogonal distribution to Fkerπin kerπby μ.Then we have

kerπ=Fkerπμ.E14

Proposition 3.1. Let M1g1Fbe an almost product Riemannian manifold and M2g2be a Riemannian manifold. Then μis invariant with respect to F.

Proof. For ZΓμand VΓkerπ, by using (2), we have g1FZFV=0,which show that FZis orthogonal to Fkerπ. On the other hand, since FVand Zare orthogonal we get g1FVZ=g1VFZ=0which shows that FZis orthogonal to kerπ.This completes proof.□

For ZΓkerπ, we have

FZ=BZ+CZ,E15

where BZΓkerπand CZΓμ.On the other hand, since πkerπ=TM2and πis a conformal submersion, using (15) we derive 1λ2g2πFVπCZ=0for any ZΓkerπand VΓkerπ,which implies that

TM2=πFkerππμ.E16

Lemma 3.1. Let πbe a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. Then we have

g1CWFV=0E17

and

g1ZM1CWFV=g1CWFAZVE18

for Z,WΓkerπand VΓkerπ.

Proof. For WΓkerπand VΓkerπ, using (2) we have

g1CWFV=g1FWBWFV=g1FWFV

due to BWΓkerπand FVΓkerπ.Hence g1FWFV=g1WV=0which is (17). Since M1is a locally product Riemannian manifold, differentiating (3.4) with respect to Z, we get

g1ZM1CWFV=g1CWFZM1V

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

g1ZM1CWFV=g1CWFAZVg1CWFVZM1V.

Since FVZM1VΓFkerπ, we obtain (18).□

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

Theorem 3.1. Let π:M1g1FM2g2is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1Fto a Riemannian manifold M2g2.Then the following assertions are equivalent to each other;

  1. kerπis integrable,

  2. 1λ2g2WππCZZππCWπFV=g1AZBWAWBZCWlnλZ+CZlnλWFVE19

for any Z,WΓkerπand VΓkerπ.

Proof. For WΓkerπand VΓkerπ, we see from Definition 3.1, FVΓkerπand FWΓkerπμ. Thus using (2) and (3), for ZΓkerπwe obtain

g1ZWV=g1ZM1FWFVg1WM1FZFV.

Further, from (15) we get

g1ZWV=g1ZM1BWFV+g1ZM1CWFVg1WM1BZFVg1WM1CZFV.

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

g1ZWV=g1AZBWAWBZFV+1λ2g2πZM1CWπFV1λ2g2πWM1CZπFV.

Thus, from (12) and Lemma 2.2 we derive

g1ZWV=g1AZBWAWBZFVg1HgradlnλZg1CWFVg1HgradlnλCWg1ZFV+g1ZCWg1HgradlnλFV+1λ2g2ZππCWπFV+g1HgradlnλWg1CZFV+g1HgradlnλCZg1WFVg1WCZg1HgradlnλFV1λ2g2WππCZπFV.

Moreover, using (17), we obtain

g1ZWV=g1AZBWAWBZCWlnλZ+CZlnλWFV1λ2g2WππCZZππCWπFV

which proves ab. □

From Theorem 3.1, we deduce the following characterization.

Theorem 3.2. Let πbe a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. Then any two conditions below imply the three;

  1. kerπis integrable.

  2. λis a constant on Γμ.

  3. g2WππCZZππCWπFV=λ2g1AZBWAWBZFV

for Z,WΓkerπand VΓkerπ.

Proof. From Theorem 3.1, we have

g1ZWV=g1AZBWAWBZCWlnλZ+CZlnλWFV1λ2g2WππCZZππCWπFV.

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

g1HgradlnλCWg1ZFV+g1HgradlnλCZg1WFV=0.E20

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

g1HgradlnλCFVg1ZFV+g1HgradlnλCZg1FVFV=0.

Hence λis 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 πbe a conformal Lagrangian submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. Then the following assertions are equivalent to each other:

  1. kerπis integrable

  2. AZFW=AWFZ

  3. πZFW=πWFZ

for Z,WΓkerπ.

Proof. For Z,WΓkerπand VΓkerπ, we see from Definition 3.1, FVΓkerπand FWΓkerπ. From Theorem 3.1 we have

g1ZWV=g1AZBWAWBZCWlnλZ+CZlnλWFV1λ2g2WππCZZππCWπFV.

Since πis 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 kerπand kerπ. For the geometry of leaves of the horizontal distribution kerπ, we have the following theorem.

Theorem 4.1. Let π:M1g1FM2g2is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1Fto a Riemannian manifold M2g2.Then the following assertions are equivalent to each other;

  1. kerπdefines a totally geodesic foliation on M1.

  2. 1λ2g2ZππCWπFV=g1AZBWCWlnλZ+g1ZCWlnλFV

for Z,WΓkerπand VΓkerπ.

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

g1ZM1WV=g1AZBWFV+g1ZM1CWFV.

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

g1ZM1WV=g1AZBWFV1λ2g1HgradlnλZg2πCWπFV1λ2g1HgradlnλCWg2πZπFV+1λ2g1ZCWg2πHgradlnλπFV+1λ2g2ZππCWπFV.

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 πbe a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. Then any two conditions below imply the three;

  1. kerπdefines a totally geodesic foliation on M1.

  2. πis horizontally homothetic submersion.

  3. g2ZππCWπFV=λ2g1AZFVBW

for Z,WΓkerπand VΓkerπ.

Proof. For Z,WΓkerπand VΓkerπ, from Theorem 4.1, we have

g1ZM1WV=g1AZBWCWlnλZ+g1ZCWlnλFV+1λ2g2ZππCWπFV.

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

g1HgradlnλCWg1ZFV+g1HgradlnλFVg1ZCW=0.E21

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

g1HgradlnλFVgMCWCW=0.

Thus, λis a constant on ΓFkerπ. On the other hand, taking Z=FVin (25) and using (17) we derive

g1HgradlnλCWg1FVFV=0.

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

For conformal Lagrangian submersion, we have the following result.

Corollary 4.1. Let πbe a conformal Lagrangian submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. Then the following assertions are equivalent to each other;

  1. kerπdefines a totally geodesic foliation on M1.

  2. AZBW=0

  3. πZFV=0

for Z,WΓkerπand VΓkerπ.

Proof. For Z,WΓkerπand VΓkerπ, from Theorem 4.1, we have

g1ZM1WV=g1AZBWCWlnλZ+g1ZCWlnλFV+1λ2g2ZππCWπFV.

Since πis 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:M1g1FM2g2is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1Fto a Riemannian manifold M2g2.Then the following assertions are equivalent to each other;

  1. kerπdefines a totally geodesic foliation on M1.

  2. 1λ2g2FUππFVπFCZ=g1TVFUBZ+g1UVg1HgradlnλFCZ

for V,UΓkerπand ZΓkerπ.

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

g1VM1UZ=g1TVFUBZ+g1HVM1FUCZ.

Since M1is torsion free and VFUΓkerπwe obtain

g1VM1UZ=g1TVFUBZ+g1FUM1VCZ.

Using (3) and (10) we have

g1VM1UZ=g1TVFUBZ+g1FUM1FVFCZ

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

g1VM1UZ=g1TVFUBZ+1λ2g1HgradlnλFUg2πFVπFCZ1λ2g1HgradlnλFVg2πFUπFCZ+g1FUFV1λ2g2πHgradlnλπFCZ+1λ2g2FUππFVπFCZ.

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

g1VM1UZ=g1TVFUBZ+g1UVg1HgradlnλFCZ+1λ2g2FUππFVπFCZ

which proves iii. □

From Theorem 4.3, we deduce the following result.

Theorem 4.4. Let πbe a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. Then any two conditions below imply the three;

  1. kerπdefines a totally geodesic foliation on M1

  2. λis a constant on Γμ

  3. 1λ2g2FUππFVπFCZ=g1TVFUBZ

for V,UΓkerπand ZΓkerπ.

Proof. For V,UΓkerπand ZΓkerπ, from Theorem 4.3 we have

g1VM1UZ=g1TVFUBZ+g1UVg1HgradlnλFCZ+1λ2g2FUππFVπFCZ.

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

g1UVg1HgradlnλFCZ=0.

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

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

Corollary 4.2. Let πbe a conformal Lagrangian submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. Then the following assertions are equivalent to each other;

  1. kerπdefines a totally geodesic foliation on M1.

  2. TVFU=0

for V,UΓkerπand ZΓkerπ.

Proof. From Theorem 4.3 we have

g1VM1UZ=g1TVFUBZ+g1UVg1HgradlnλFCZ+1λ2g2FUππFVπFCZ.

for V,UΓkerπand ZΓkerπ. Since πis 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 πbetween two Riemannian manifolds is called totally geodesic if π=0[15].

Theorem 5.1. Let π:M1g1FM2g2is a conformal anti-invariant submersion from an almost product Riemannian manifold M1g1Fto a Riemannian manifold M2g2.πis totally geodesic map if and only if.

  1. πis a horizontally homothetic map,

  2. TUFV=0and HUM1FVΓFkerπ,

  3. AZFV=0and HZM1FVΓFkerπ

for Z,W,ZΓkerπand U,VΓkerπ.

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

πZW=ZlnλπW+WlnλπZg1ZWπgradlnλ.

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

ZlnλπFZ+FZlnλπZg1ZFZπgradlnλ=0.E22

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

g1gradlnλZλ2g1FZFZ+g1gradlnλFZλ2g1ZFZg1ZFZλ2g1gradlnλFZ=0.E23

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

πFUFV=FUlnλπFV+FVlnλπFUg1FUFVπgradlnλ.

Again if πis a horizontally homothetic map, then πFUFV=0.Conversely, if πFUFV=0, putting Uinstead of Vin above equation, we derive

2FUlnλπFUg1FUFUπgradlnλ=0.E24

Taking inner product in (33) with πFUand since πis a conformal submersion, we have

g1FUFUλ2g1gradlnλFU=0.

From above equation, λis a constant on ΓFkerπ.Thus λis 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=0if and only if

πFTUFV+CHUM1FV=0E25

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

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

πZV=ZππVπZM1V=πFZM1FV.

Using (9) and (10) we have

πZV=πFAZFV+CHZM1FV.

Thus πZV=0if and only if

πFAZFV+CHZM1FV=0.

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

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

Theorem 5.2 Let πbe a conformal anti-invariant submersion from a locally product Riemannian manifold M1g1Fonto a Riemannian manifold M2g2. If πis a totally geodesic map then

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

for any ZΓkerπand W=W1+W2ΓTM,where W1Γkerπand W2Γkerπ.

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

πZW=ZππWπFZM1FW

for any ZΓkerπand 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 W1Γkerπand W2Γkerπ. 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 Iis 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 R4with coordinates x1x4, we can canonically choose an almost product structure Fon R4as follows:

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

Example 6.2. Let πbe a submersion defined by

π:R4x1x2x3x4R2cosx1sinhx2sinx1coshx2.

Then it follows that

kerπ=spanV1=x3V2=x4

and

kerπ=spanX1=x1X2=x2.

Hence, we have FV1=X1and FV2=X2imply 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 g1and g2denote the standard metrics (inner products) of R4and R2. Thus πis a conformal anti-invariant submersion with λ2=sin2x1sinh2x2+cos2x1cosh2x2.

Example 6.3. Let πbe a submersion defined by

π:R4x1x2x3x4R2ex3sinx42ex3cosx42.

Then it follows that

kerπ=spanV1=x1V2=x2

and

kerπ=spanW1=x3W2=x4.

Hence we have FV1=W1and FV2=W2imply that Fkerπ=kerπ.Also by direct computations, we get

πW1=ex3sinx42y1+ex3cosx42y2,πW2=ex3cosx42y1ex3sinx42y2.

Hence, we have

g2πW1πW1=ex322g1W1W1,g2πW2πW2=ex322g1W2W2,

where g1and g2denote the standard metrics (inner products) of R4and R2. Thus πis a conformal anti-invariant submersion with λ=ex32.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mehmet Akif Akyol (April 30th 2019). On Conformal Anti-Invariant Submersions Whose Total Manifolds Are Locally Product Riemannian, Manifolds II - Theory and Applications, Paul Bracken, IntechOpen, DOI: 10.5772/intechopen.80337. Available from:

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