Open access peer-reviewed chapter - ONLINE FIRST

Clairaut Submersion

By Sanjay Kumar Singh and Punam Gupta

Reviewed: October 28th 2021Published: December 24th 2021

DOI: 10.5772/intechopen.101427

Downloaded: 23

Abstract

In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion.

Keywords

  • Riemannian submersion
  • nearly Kähler manifolds
  • Kähler manifolds
  • anti-invariant submersion
  • semi-invariant submersion
  • clairaut submersion
  • totally geodesic maps

1. Introduction

Riemannian submersion between two Riemannian manifolds was first introduced by O’Neill [1] and Gray [2]. After that Watson [3] introduced almost Hermitian submersions. Later, the notion of anti-invariant submersions and Lagrangian submersion from almost Hermitian manifolds onto Riemannian manifolds were introduced by Sahin [4] and studied by Taştan [5, 6], Gündüzalp [7], Beri et al. [8], Ali and Fatima [9], in which the fibers of submersion are anti-invariant with respect to the almost complex structure of total manifold. After that several new types of Riemannian submersions were defined and studied such as semi-invariant submersion [10, 11], slant submersion [12, 13], generic submersion [14, 15, 16, 17], hemi-slant submersion [18], semi-slant submersion [19], pointwise slant submersion [20, 21, 22] and conformal semi-slant submersion [23]. Also, these kinds of submersions were considered in different kinds of structures such as nearly Kähler, Kähler, almost product, para-contact, Sasakian, Kenmotsu, cosymplectic and etc. In book [24], we find the recent developments in this field.

In 1735, A.C. Clairaut [25] obtained the very important result in the theory of surfaces, which is Clairaut’s theorem and stated that for any geodesic αon a surface of revolution S, the function rsinθis constant along α, where ris the distant from a point on the surface to the rotation axis and θis the angle between αand the meridian through α. Bishop [26] introduced the idea of Riemannian submersions and gave a necessary and sufficient conditions for a Riemannian submersion to be Clairaut. Allison [27] considered Clairaut semi-Riemannian submersions and showed that such submersions have interesting applications in the static space-times.

In [28], Tastan and Gerdan gave new Clairaut conditions for anti-invariant submersions whose total manifolds are Sasakian and Kenmotsu and got many interesting results. In [29], Tastan and Aydin studied Clairaut anti-invariant submersions whose total manifolds are cosymplectic. Gündüzalp [30] introduced Clairaut anti-invariant submersions from a paracosymplectic manifold and gave characterization theorems. In [31], Lee et al. studied Clairaut anti-invariant submersions whose total manifolds are Kähler.

Kähler manifolds [32, 33] have an especially rich geometric and topological structure because of Kähler identity. Kähler manifolds are very important in differential geometry, which has applications in several different fields such as supersymmetric gauge theory and superstring theory in theoretical physics, signal processing in information geometry. The simplest example of Kähler manifold is a complex Euclidean space nwith the standard Hermitian metric.

Nearly Kähler manifolds introduced by Gray and Hervella [32], are the geometrically interesting class among the sixteen classes of almost Hermitian manifolds. The geometrical meaning of nearly Kähler condition is that the geodesics on the manifolds are holomorphically planar curves. Gray [2] studied nearly Kähler manifolds broadly and gave example of a non-Kählerian nearly Kähler manifold, which is 6-dimensional sphere.

Motivated by this, the authors [34] studied Clairaut anti-invariant submersions from nearly Kähler manifolds onto Riemannian manifolds with some examples and obtained conditions for Clairaut Riemannian submersion to be totally geodesic map. The authors investigated conditions for the Clairaut anti-invariant submersions to be a totally umbilical map. The authors [34] studied Clairaut semi-invariant submersions from Kähler manifolds onto Riemannian manifolds with some examples. The authors also obtained conditions for Clairaut semi-invariant Riemannian submersion to be totally geodesic map and investigated conditions for the semi-invariant submersion to be a Clairaut map.

Advertisement

2. Almost complex manifold

An almost complex structure on a smooth manifold Mis a smooth tensor field φof type 11such that φ2=I. A smooth manifold equipped with such an almost complex structure is called an almost complex manifold. An almost complex manifold Mφendowed with a chosen Riemannian metric gsatisfying

gφXφY=gXYE1

for all X,YTM, is called an almost Hermitian manifold.

An almost Hermitian manifold Mis called a nearly Kähler manifold [2] if

XφY+YφX=0E2

for all X,YTM. If XφY=0for all X,YTM, then Mis known as Kähler manifold [33]. Every Kähler manifold is nearly Kähler but converse need not be true.

Advertisement

3. Riemannian submersion

Definition 1.1 [1, 35] Let Mgmand Ngnbe Riemannian manifolds, where dimM=m, dimN=nand m>n. A Riemannian submersion π:MNis a map of Monto Nsatisfying the following axioms:

  1. πhas maximal rank.

  2. The differential πpreserves the lengths of horizontal vectors.

For each qN, π1qis an mn-dimensional Riemannian submanifold of M. The submanifolds π1q, qN, are called fibers. A vector field on Mis called vertical if it is always tangent to fibers. A vector field on Mis called horizontal if it is always orthogonal to fibers. A vector field Xon Mis called basic if Xis horizontal and π-related to a vector field X'on N, that is, πXp=Xπpfor all pM.We denote the projection morphisms on the distributions kerπand kerπby Vand H, respectively. The sections of Vand Hare called the vertical vector fields and horizontal vector fields, respectively. So

Vp=Tpπ1q,Hp=Tpπ1q.

The second fundamental tensors of all fibers π1q,qNgives rise to tensor field Tand Ain Mdefined by O’Neill [1] for arbitrary vector field Eand F, which is

TEF=HVEMVF+VVEMHF,E3
AEF=HHEMVF+VHEMHF,E4

where Vand Hare the vertical and horizontal projections.

To discuss geodesics, we need a linear connection. We denote the Levi-Civita connection on Mby ̂and the adapted connection of the submersion by . From Eqs. (3) and (4), we have

VW=TVW+̂VW,E5
VX=HVX+TVX,E6
XV=AXV+VXV,E7
XY=HXY+AXY,E8

for all V,WΓkerπand X,YΓkerπ,where VVW=̂VW.If Xis basic, then AXV=HVX.

It is easily seen that for pM,UVpand XHpthe linear operators

TU,AX:TpMTpM

are skew-symmetric, that is,

gAXEF=gEAXFandgTUEF=gETUF,E9

for all E,FTpM.We also see that the restriction of Tto the vertical distribution Tkerπ×kerπis exactly the second fundamental form of the fibers of π. Since TUis skew-symmetric, therefore πhas totally geodesic fibers if and only if T0.

Let π:MgmNgnbe a smooth map between Riemannian manifolds. Then the differential πof πcan be observed as a section of the bundle HomTMπ1TNM, where π1TNis the bundle which has fibers π1TNx=TfxN. HomTMπ1TNhas a connection induced from the Riemannian connection Mand the pullback connection N[36, 37]. Then the second fundamental form of πis given by

πEF=ENπFπEMF,forallE,FΓTM.E10

We also know that πis said to be totally geodesic map [36] if πEF=0,for all E,FΓTM.

Advertisement

4. Clairaut submersion from Riemannian manifold

Let Sbe a revolution surface in R3with rotation axis L. For any pS, we denote by rpthe distance from pto L. Given a geodesic α:JRSon S, let θtbe the angle between αtand the meridian curve through αt,tI. A well-known Clairaut’s theorem [25] named after Alexis Claude de Clairaut, says that for any geodesic on S, the product rsinθis constant along α, i.e., it is independent of t. For proof, see [38, p.183]. In the theory of Riemannian submersions, Bishop [26] introduced the notion of Clairaut submersion in the following way:

Definition 1.2 [26] A Riemannian submersion π:MgNgnis called a Clairaut submersion if there exists a positive function ron M, which is known as the girth of the submersion, such that, for any geodesic αon M, the function rαsinθis constant, where, for any t,θtis the angle between α̇tand the horizontal space at αt.

For further use, we are stating one important result of Bishop.

Theorem 1.1 [26] A curve hin Mis a geodesic if and only if Ẋ+2AXU+TUU=0and EU+TUX=0,where ḣt=E=X+U, Xis horizontal and Uis vertical.

Bishop also gave the following necessary and sufficient condition for a Riemannian submersion to be a Clairaut submersion, which is

Theorem 1.2 [26] Let π:MgNgnbe a Riemannian submersion with connected fibers. Then, πis a Clairaut submersion with r=efif and only if each fiber is totally umbilical and has the mean curvature vector field H=gradf, where gradfis the gradient of the function fwith respect to g.

Proof:Let π:MNbe a Riemannian submersion. For a geodesic hin M, we use ḣs=E=X+U, where Xis horizontal and Uis vertical. and =ḣs2. Let θsbe the angle between ḣsand the horizontal space at hs. Then

gXsXs=cos2θs,E11
gUsUs=sin2θs.E12

Differentiating (12), we get

gḣsUsUs=sinθscosθssds.E13

Using Theorem 1.1, (13) becomes

gTUsXsUs=sinθscosθssds.E14

Since TUis skew-symmetric, so form above equation, we have

gTUsUsXs=sinθscosθssds.E15

Now, πis a Clairaut submersion with r=efif and only if ddsefhsinθ=0.

Using (12, 15) in ddsefhsinθ=0, we have

gUsUsddsfhs+gTUsUsXs=0,E16
gUsUsggradfEs+gTUsUsXs=0.E17

Consider any geodesic hon Mwith initial vertical tangent vector, so gradfturns out to be horizontal. Therefore, the function fis constant on any fiber, the fibers being connected. Therefore (17) reduces to

gUsUsggradfXs+gTUsUsXs=0,E18
gUsUsgradf+TUsUs=0.E19

Setting U=U1+U2,where U1,U2are vertical vector fields and using the fact that Tis symmetric for vertical vector fields, we obtain

gU1U2gradf+TU1U2=0E20

holds for all vertical vector fields U1,U2..

Since the restriction of Tto the vertical distribution Tkerπ×kerπis exactly the second fundamental form of the fibers of π.It means that any fiber is totally umbilical with mean curvature vector field H=gradf.

Conversely, suppose the fibers are totally umbilic with normal curvature vector field H=gradfso that we have

gUUH+TUU=0.E21

Since gradfis orthogonal to fibers, so

gUUggradfE=gUUgHX=gTUUX.E22

Since (18) holds. so rhsinθis constant along any geodesic h.

Example 1.1 [24] Consider the warped product manifold M1×fM2of Riemannian manifolds M1g1and M2g2, where f:M10. The fibers of the first projection p1:M1×fM2M1are totally umbilical with mean curvature vector field H=gradlogf1/2. Thus, if M2is connected, p1is a Clairaut submersion with r=f1/2.

Advertisement

5. Anti-invariant Riemannian submersion

Definition 1.3 [39] Let Mφgbe an almost Hermitian manifold and Nbe a Riemannian manifold with Riemannian metric gn. Suppose that there exists a Riemannian submersion π:MN, such that the vertical distribution kerπis anti-invariant with respect to φ, i.e., φkerπkerπ. Then, the Riemannian submersion πis called an anti-invariant Riemannian submersion. We will briefly call such submersions as anti-invariant submersions.

Let πbe an anti-invariant Riemannian submersion from nearly Kähler manifold Mφgmonto Riemannian manifold Ngn. For any arbitrary tangent vector fields Uand Von M, we set

UφV=PUV+QUVE23

where PUV,QUVdenote the horizontal and vertical part of UφV, respectively. Clearly, if Mis a Kähler manifold then P=Q=0.

If Mis a nearly Kähler manifold then Pand Qsatisfy

PUV=PVU,QUV=QVU.E24

Consider

kerπ=φkerπμ,

where μis the complementary distribution to φkerπin kerπand φμμ.

For XΓkerπ, we have

φX=αX+βX,E25

where αXΓkerπand βXΓμ. If μ=0, then an anti-invariant submersion is known as Lagrangian submersion.

5.1 Anti-invariant Clairaut submersions from nearly Kähler manifolds

In this section, we give new Clairaut conditions for anti-invariant submersions from nearly Kähler manifolds after giving some auxiliary results.

Theorem 1.3 [34] Let πbe an anti-invariant submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngn. If h:JRMis a regular curve and Usand Xsare the vertical and horizontal parts of the tangent vector field ḣs=Wof hs, respectively, then his a geodesic if and only if along h

AXφU+AXβX+TUβX+VXαX+TUφU+̂UαX=0,E26
HḣφU+ḣβX+AXαX+TUαX=0.E27

Proof:Let πbe an anti-invariant submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngn. Since φ2ḣ=ḣ. Taking the covariant derivative of this and using (2), we have

ḣφφḣ+φḣφḣ=ḣḣ.E28

Since Usand Xsare the vertical and horizontal parts of the tangent vector field ḣs=Wof hs, that is, ḣ=U+X. So (28) becomes

ḣḣ=φU+XφU+X+Pḣφḣ+Qḣφḣ=φUφU+XφU+UφX+XφX+Pḣφḣ+Qḣφḣ=φUφU+XφU+UαX+βX+XαX+βX+Pḣφḣ+Qḣφḣ.E29

Using (5)(8) in (29), we get

ḣḣ=φHḣφU+ḣβX+AXαX+AXβX+AXφU+TUβX+TUαX+VXαX+TUφU+̂UαX+Pḣφḣ+Qḣφḣ.E30

Let Y,ZTM. Since φ2Z=Z, on differentiation, we have

φYφZ+YφφZ=YZ,
φ2YZ+φYφZ+YφφZ=YZ,

using (23) in above, we obtain

φPYZ+QYZ=PYφZQYφZ.E31

By (31), we have

φPḣφḣ+Qḣφḣ=Pḣḣ+Qḣḣ,

since Pand Qare skew-symmetric, so

φPḣφḣ+Qḣφḣ=0.E32

Using (32) and equating the vertical and horizontal part of (30), we obtain

Vφḣḣ=AXφU+AXβX+TUβX+VXαX+TUφU+̂UαX,
Hφḣḣ=HḣφU+ḣβX+AXαX+TUαX.

By using above equations, we can say that his geodesic if and only if (26, 27) hold.

Theorem 1.4 [34] Let πbe an anti-invariant submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngn. Also, let h:JRMbe a regular curve and Usand Xsare the vertical and horizontal parts of the tangent vector field ḣs=Wof hs. Then πis a Clairaut submersion with r=efif and only if along h

ggradfXgUU=gHḣβX+AXαX+TUαX+PḣsUφU.

Proof:Let h:JRMbe a geodesic on Mand =ḣs2. Let θsbe the angle between ḣsand the horizontal space at hs. Then

gXsXs=cos2θs,E33
gUsUs=sin2θs.E34

Differentiating (34), we get

2gḣsUsUs=2sinθscosθssds.E35

Using (1) in (35), we have

gHḣsφUsφUsgḣsφUsφUs=sinθscosθssds.

Now by use of (23), we have

gHḣsφUsφUsgPḣsU+QḣsUφUs=sinθscosθssds.

Along the curve h, using Theorem 1.3, we obtain

gHḣβX+AXαX+TUαX+PḣsUφUs=sinθscosθssds.

Now, πis a Clairaut submersion with r=efif and only if ddsefsinθ=0. Therefore

efdfdssinθ+cosθds=0,efdfdssin2θ+sinθcosθds=0.

So, we obtain

dfdshsgUsUs=gHḣβX+AXαX+TUαX+PḣsUφUs.E36

Since dfdshs=ggradfḣs=ggradfX. Therefore by using (36), we get the result.

Theorem 1.5 [34] Let πbe an Clairaut anti-invariant submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. Then

AφWφX+QWφX=XfW

for Xkerπ, Wkerπand φWis basic.

Proof:Let πbe an anti-invariant submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. We know that any fiber of Riemannian submersion πis totally umbilical if and only if

TVW=gVWH,E37

for all V,WΓkerπ, where Hdenotes the mean curvature vector field of any fiber in M. By using Theorem 1.2 and (37), we have

TVW=gVWgradf.E38

Let Xμand V,WΓkerπ, then by using (1) and (2), we have

gVφWφX=gφVW+VφWφX=gVWX+gPVW+QVWφX.E39

By using (1), we have

gφYZ=gYφZ,

where Y,ZTM. Taking covariant derivative of above, we get

gXφYZ=gYXφZ,

using (23), we get

gPXY+QXYZ=gYPXZ+QXZ=gYPZX+QZX.E40

Using (40), we have

gPWφX+QWφXV=gφXPVW+QVW.E41

Using (5), (38), (41) in (39), we have

gVφWφX=gVWgradfX+gVQWφX.

Since φWis basic, so HVφW=AφWV, therefore we have

gAφWVφX=gVWgradfX+gVQWφX,
gVAφWφX+gVQWφX=gVWgradfXE42

because Ais skew-symmetric. By using (42), we get the result.

Theorem 1.6 [34] Let πbe a Clairaut anti-invariant submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=efand gradfφkerπ. Then either fis constant on φkerπor the fibers of πare 1-dimensional.

Proof:Using (5) and (38), we have

gVWφU=gVWggradfφU,

where U,V,WΓkerπ. Since gWφU=0. therefore we have

gWVφU=gVWggradfφU.E43

By use of (1) and (23) in (43), we get

gWQVUgφWVU=gVWggradfφU.

By using (5), we obtain

gWQVUgφWTVU=gVWggradfφU.

Now, using (38), we get

gWQVU+gVUggradfφW=gVWggradfφUE44

Take V=Uin (44), we have

gVVggradfφW=gVWggradfφV.E45

Interchange Vwith Win (45), we have

gWWggradfφV=gVWggradfφW.E46

By (45) and (46), we have

g2VWggradfφV=gVVgWWggradfφV.

Therefore either fis constant on φkerπor V=aW, where ais constant (by using Schwarz’s Inequality for equality case).

Corollary 1.1 [34] Let πbe a Clairaut anti-invariant submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=efand gradfφkerπ. If dimkerπ>1, then the fibers of πare totally geodesic if and only if AφWφX+QWφX=0for Wkerπsuch that φWis basic and Xμ.

Proof:By Theorem 1.5 and Theorem 1.6, we get the result.

Corollary 1.2 [34] Let πbe an Clairaut Lagrangian submersion from a nearly Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. Then either the fibers of πare 1-dimensional or they are totally geodesic.

Proof:Let πbe an Clairaut Lagrangian submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. Then μ=0, so AφWφX+QWφX=0always.

Now, we discuss some examples for Clairaut anti-invariant submersions from a nearly Kähler manifold.

Example 1.2 [34] Let R4φgbe a nearly Kähler manifold endowed with Euclidean metric gon R4given by

g=i=14dxi2

and canonical complex structure

φxj=xj+1j=1,3xj1j=2,4.

The φ-basis is ei=xii=1,2,3,4. Let R3g1be a Riemannian manifold endowed with metric g1=i=13dyi2.

  1. Consider a map π:R4φgR3g1defined by

πx1x2x3x4=x1+x22x3x4.

Then by direct calculations, we have

kerπ=spanX1=x1x2,kerπ=spanX2=x1+x2X3=x3X4=x4

and φX1=X2, therefore φkerπkerπ. Thus, we can say that πis an anti-invariant Riemannian submersion. Since the fibers of πare 1-dimensional, therefore fibers are totally umbilical.

Consider the Koszul formula for Levi-Civita connection for R4

2gXYZ=XgYZ+YgZXZgXYgYZXgXZY+gXYZ

for all X,Y,ZR4. By simple calculations, we obtain

eiej=0foralli,j=1,2,3,4.

Hence TXY=TYX=TXX=0for all X,YΓkerπ. Therefore fibers of πare totally geodesic. Thus πis Clairaut trivially.

  1.   ii. Consider a map π:R4φgR3g1defined by

πx1x2x3x4=x12+x22x3x4.

Then by direct calculations, we have

kerπ=spanX1=x2x12+x22x1x1x12+x22x2,kerπ=spanX2=x2x12+x22x1+x1x12+x22x2X3=x3X4=x4

and φX1=X2, therefore φkerπkerπ. Thus, we can say that πis an anti-invariant Riemannian submersion. Since the fibers of πare 1-dimensional, therefore fibers are totally umbilical. By using Koszul formula, we obtain

eiej=0foralli,j=1,2,3,4.

Hence

TX1X1=x2x12+x22x1+x1x12+x22x2.

Now, for the function f=lnx12+x22on R4φg, the gradient of fwith respect to gis given by

gradf=i,j=14gijfxixj=x2x12+x22x1+x1x12+x22x2.

Therefore for X1Γkerπ, TX1X1=gradf. Since X1=1, so TX1X1=X12gradf. By using Theorem 1.2, we can say that πis an proper Clairaut anti-invariant submersion with r=effor f=lnx12+x22.

Remark:From all results of this section, we can easily find conditions for anti-invariant Clairaut Submersions from Kähler manifolds.

Advertisement

6. Semi-invariant Riemannian submersion

Definition 1.4 Let Mφgbe an almost Hermitian manifold and Nbe a Riemannian manifold with Riemannian metric gn. A Riemannian submersion π:MNis called a semi-invariant Riemannian submersion [11] if there is a distribution D1kerπsuch that

kerπ=D1D2andφD1=D1,φD2kerπ,

where D2is orthogonal complementary to D1in kerπ. For VΓkerπ, we have

φV=ϕV+ωV,E47

where ϕVΓD1and ωVΓφD2.

Definition 1.5 A semi-invariant Riemannian submersion πis said to be a Lagrangian Riemannian submersion [4] if φkerπ=kerπ. Hence, if πis a Lagrangian Riemannian submersion then for any VΓkerπ,φV=ωV,ϕV=0and for XΓkerπ, φX=αX, βX=0.

6.1 Semi-invariant Clairaut submersions from Kähler manifolds

In this section, we give new Clairaut conditions for semi-invariant submersions from Kähler manifolds after giving some auxiliary results.

Theorem 1.7 [40] Let πbe a semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngn. If h:JRMis a regular curve and Usand Xsare the vertical and horizontal parts of the tangent vector field ḣs=Wof hs, respectively, then his a geodesic if and only if along h

VXϕU+VXαX+AXωU+AXβX+̂UϕU+TUβX+TUωU+̂UαX=0,E48
AXϕU+AXαX+HḣωU+ḣβX+TUϕU+TUαX=0.E49

Proof:Let πbe a semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngn. Since φ2ḣ=ḣ. Taking the covariant derivative of this and using (2), we have

φḣφḣ=ḣḣ.E50

Since Usand Xsare the vertical and horizontal parts of the tangent vector field ḣs=Wof hs, that is, ḣ=U+X. So (50) becomes

ḣḣ=φU+XφU+X=φUφU+XφU+UφX+XφX=φUϕU+ωU+XϕU+ωU+UαX+βX+XαX+βX.E51

Using (5)(8) in (51), we get

ḣḣ=φHḣφU+ḣβX+AXαX+AXβX+AXφU+TUβX+TUαX+VXαX+TUφU+̂UαX.E52

Equating the vertical and horizontal part of (52), we obtain

Vφḣḣ=VXϕU+VXαX+AXωU+AXβX+̂UϕU+TUβX+TUωU+̂UαX,
Hφḣḣ=AXϕU+AXαX+HḣωU+ḣβX+TUϕU+TUαX.

By using above equations, we can say that his geodesic if and only if (48) and (49) hold.

Theorem 1.8 [40] Let πbe a semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngn. Also, let h:JRMbe a regular curve. Usand Xsare the vertical and horizontal parts of the tangent vector field ḣs=Wof hs. Then πis a Clairaut submersion with r=efif and only if along h

ggradfXgUU=gHḣβX+AXαX+TUαXωU+gVXαX+AXβX+TUβX+̂UαXϕU.

Proof:Let h:JRMbe a geodesic on Mand =ḣs2. Let θsbe the angle between ḣsand the horizontal space at hs. Then

gXsXs=cos2θs,E53
gUsUs=sin2θs.E54

Differentiating (54), we get

2gḣsUsUs=2sinθscosθssds.E55

Using (1) in (55), we have

gḣsφUsφUs)=sinθscosθssds.

Now by use of (47), we have

gḣsϕUsφUs+gḣsωUsφUs=sinθscosθssds

Along the curve h, using Theorem 1.7 and (5)(8), we obtain

sinθscosθssds=gHḣβX+AXαX+TUαXωUsgVXαX+AXβX+TUβX+̂UαXϕUs

Now, πis a Clairaut submersion with r=efif and only if ddsefsinθ=0. Therefore

efdfdssinθ+cosθds=0,efdfdssin2θ+sinθcosθds=0.

So, we obtain

dfdshsgUsUs=gHḣβX+AXαX+TUαXωUs+gVXαX+AXβX+TUβX+̂UαXϕUs,E56

Since dfdshs=ḣgs=ggradfḣs=ggradfX. Therefore by using (56), we get the result.

Theorem 1.9 [40] Let πbe a Clairaut semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. Then

gAωWVβX+gHVϕWβX+gVVωWαX+ĝVϕWαX=gVWXf

for XΓμ, V,WΓD2and ωWis basic.

Proof:Let πbe a Clairaut semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. We know that any fiber of Riemannian submersion πis totally umbilical if and only if

TVW=gVWH,E57

for all V,WΓkerπ, where Hdenotes the mean curvature vector field of any fiber in M. By using Theorem 1.2 and (57), we have

TVW=gVWgradf.E58

Let Xμand V,WΓkerπ, then by using (1) and (2), we have

gVφWφX=gφVW+VφWφX=gVWX.E59

Using (5), (58) in (59), we have

gVφWφX=gVWgradfX.

Since ωWis basic, so HVωW=AωWV, therefore we have

gAωWVβX+gHVϕWβX+gVVωWαX+ĝVϕWαX=gVWgradfX.E60

Theorem 1.10 [40] Let πbe a Clairaut semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=efand V,WΓD1Then gradfΓφkerπ.

Proof:Let V,WΓD1and XΓμ. Using (5), (47) and (58) in

VφU=φVU+VφU,

we have

TVϕU+VVϕU=αTVU+βTVU+ϕVVU+ωVVU,

which gives

gVϕUggradfX=gVUggradfφX.E61

By interchanging Uand Vin (61) and adding the resulting equation with (61), we get

gVUggradfφX=0,

which gives ggradfφX=0. Therefore gradfΓφkerπ.

Theorem 1.11 [40] Let πbe a Clairaut semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=efand gradfΓφkerπ. Then either fis constant on φkerπor the fibers of πare 1-dimensional.

Proof:Let U,VΓD2. Using (5) and (58), we have

gVUφU=gVUggradfφU,

which gives

gφVUU=gVUggradfφU,

since kerπis integrable, so we have

gφUVU=gVUggradfφU,

which equals to

gUφVUφVU=gVUggradfφU.

Since gφVU=0. therefore we have

gφVUU=gVUggradfφU.E62

By using (5) in (62), we obtain

gφVTUU=gVUggradfφU.

Now, using (58), we get

gUUggradfφV=gVUggradfφU.E63

Interchanging Vand Uin (63), we have.

gVVggradfφU=gVUggradfφV.E64

By (63) and (64), we have

g2VUggradfφU=gVVgUUggradfφU.

Therefore either fis constant on φkerπor V=aU, where ais constant (by using Schwarz’s Inequality for equality case).

Since kerπis CR-submanifold of Kähler manifold Mφg, therefore by using [41], Theorem 6.1, p. 96], we can state that.

Theorem 1.12 [40] Let πbe a Clairaut semi-invariant submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. If dim D2>1, then fibers are totally geodesic.

Corollary 1.3 [40] Let πbe a Clairaut Lagrangian submersion from a Kähler manifold Mφgonto a Riemannian manifold Ngnwith r=ef. Then either the fibers of πare 1-dimensional or they are totally geodesic.

Lastly, we discuss some examples for Clairaut semi-invariant submersions [40] from a Kähler manifold.

Example 1.3 Every Clairaut anti-invariant submersion from a Kähler manifold onto a Riemannian manifold is a Clairaut semi-invariant submersion with D1=0.

Example 1.4 Let R6φgbe a Kähler manifold endowed with Euclidean metric gon R6given by

g=i=16dxi2

and canonical complex structure

φxj=xj+1j=1,3,5xj1j=2,4,6.

The φ-basis is ei=xii=16. Let R3g1be a Riemannian manifold endowed with metric g1=i=13dyi2.

Consider a map π:R6φgR3g1defined by

πx1x2x3x4x5x6=x1+x22x3x4.

Then by direct calculations, we have

kerπ=spanX1=x1x2X2=x5X3=x6,kerπ=spanV1=x1+x2V2=x3V3=x4

and φX1=V1, φX2=X3, φX3=X2therefore D1=spanX2X3and D2=spanX1. Thus, we can say that πis a semi-invariant Riemannian submersion.

Consider the Koszul formula for Levi-Civita connection for R6

2gXYZ=XgYZ+YgZXZgXYgYZXgXZY+gXYZ

for all X,Y,ZR6.By simple calculations, we obtain

eiej=0foralli,j=1,,6.

Hence TXY=TYX=TXX=0for all X,YΓkerπ. Therefore fibers of πare totally geodesic. Thus πis Clairaut trivially.

Advertisement

Classification

2010 mathematics subject classification. 53C12, 53C15, 53C20, 53C55

DOWNLOAD FOR FREE

chapter PDF

© 2021 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.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Sanjay Kumar Singh and Punam Gupta (December 24th 2021). Clairaut Submersion [Online First], IntechOpen, DOI: 10.5772/intechopen.101427. Available from:

chapter statistics

23total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us