Open access peer-reviewed chapter

Clairaut Submersion

Written By

Sanjay Kumar Singh and Punam Gupta

Reviewed: 28 October 2021 Published: 24 December 2021

DOI: 10.5772/intechopen.101427

From the Edited Volume

Advanced Topics of Topology

Edited by Francisco Bulnes

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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 α on a surface of revolution S, the function rsinθ is constant along α, where r is the distant from a point on the surface to the rotation axis and θ is the angle between α and 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 n with 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.

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2. Almost complex manifold

An almost complex structure on a smooth manifold M is a smooth tensor field φ of type 11 such that φ2=I. A smooth manifold equipped with such an almost complex structure is called an almost complex manifold. An almost complex manifold Mφ endowed with a chosen Riemannian metric g satisfying

gφXφY=gXYE1

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

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

XφY+YφX=0E2

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

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3. Riemannian submersion

Definition 1.1 [1, 35] Let Mgm and Ngn be Riemannian manifolds, where dimM=m, dimN=n and m>n. A Riemannian submersion π:MN is a map of M onto N satisfying the following axioms:

  1. π has maximal rank.

  2. The differential π preserves the lengths of horizontal vectors.

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

Vp=Tpπ1q,Hp=Tpπ1q.

The second fundamental tensors of all fibers π1q,qN gives rise to tensor field T and A in M defined by O’Neill [1] for arbitrary vector field E and F, which is

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

where V and H are the vertical and horizontal projections.

To discuss geodesics, we need a linear connection. We denote the Levi-Civita connection on M by ̂ 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,WΓkerπ and X,YΓkerπ, where VVW=̂VW. If X is basic, then AXV=HVX.

It is easily seen that for pM, UVp and XHp the linear operators

TU,AX:TpMTpM

are skew-symmetric, that is,

gAXEF=gEAXFandgTUEF=gETUF,E9

for all E,F TpM. We also see that the restriction of T to the vertical distribution Tkerπ×kerπ is exactly the second fundamental form of the fibers of π. Since TU is skew-symmetric, therefore π has totally geodesic fibers if and only if T0.

Let π:MgmNgn be a smooth map between Riemannian manifolds. Then the differential π of π can be observed as a section of the bundle HomTMπ1TNM, where π1TN is the bundle which has fibers π1TNx=TfxN. HomTMπ1TN has a connection induced from the Riemannian connection M and the pullback connection N [36, 37]. Then the second fundamental form of π is given by

πEF=ENπFπEMF,forallE,FΓTM.E10

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

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4. Clairaut submersion from Riemannian manifold

Let S be a revolution surface in R3 with rotation axis L. For any pS, we denote by rp the distance from p to L. Given a geodesic α:JRS on S, let θt be the angle between αt and 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 rsinθ is 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:MgNgn is called a Clairaut submersion if there exists a positive function r on M, which is known as the girth of the submersion, such that, for any geodesic α on M, the function rαsinθ is constant, where, for any t,θt is the angle between α̇t and the horizontal space at αt.

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

Theorem 1.1 [26] A curve h in M is a geodesic if and only if Ẋ+2AXU+TUU=0 and EU+TUX=0, where ḣt=E=X+U, X is horizontal and U is 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:MgNgn be a Riemannian submersion with connected fibers. Then, π is a Clairaut submersion with r=ef if and only if each fiber is totally umbilical and has the mean curvature vector field H=gradf, where gradf is the gradient of the function f with respect to g.

Proof: Let π:MN be a Riemannian submersion. For a geodesic h in M, we use ḣs=E=X+U, where X is horizontal and U is vertical. and =ḣs2. Let θs be the angle between ḣs and 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 TU is skew-symmetric, so form above equation, we have

gTUsUsXs=sinθscosθssds.E15

Now, π is a Clairaut submersion with r=ef if and only if ddsefhsinθ=0.

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

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

Consider any geodesic h on M with initial vertical tangent vector, so gradf turns out to be horizontal. Therefore, the function f is 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,U2 are vertical vector fields and using the fact that T is symmetric for vertical vector fields, we obtain

gU1U2gradf+TU1U2=0E20

holds for all vertical vector fields U1,U2..

Since the restriction of T to the vertical distribution Tkerπ×kerπ is 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=gradf so that we have

gUUH+TUU=0.E21

Since gradf is orthogonal to fibers, so

gUUggradfE=gUUgHX=gTUUX.E22

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

Example 1.1 [24] Consider the warped product manifold M1×fM2 of Riemannian manifolds M1g1 and M2g2, where f:M10. The fibers of the first projection p1:M1×fM2M1 are totally umbilical with mean curvature vector field H=gradlogf1/2. Thus, if M2 is connected, p1 is a Clairaut submersion with r=f1/2.

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5. Anti-invariant Riemannian submersion

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

Let π be an anti-invariant Riemannian submersion from nearly Kähler manifold Mφgm onto Riemannian manifold Ngn. For any arbitrary tangent vector fields U and V on M, we set

UφV=PUV+QUVE23

where PUV,QUV denote the horizontal and vertical part of UφV, respectively. Clearly, if M is a Kähler manifold then P=Q=0.

If M is a nearly Kähler manifold then P and Q satisfy

PUV=PVU,QUV=QVU.E24

Consider

kerπ=φkerπμ,

where μ is the complementary distribution to φkerπ in kerπ and φμμ.

For XΓkerπ, we have

φX=αX+βX,E25

where αXΓkerπ and 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 π be an anti-invariant submersion from a nearly Kähler manifold Mφg onto a Riemannian manifold Ngn. If h:JRM is a regular curve and Us and Xs are the vertical and horizontal parts of the tangent vector field ḣs=W of hs, respectively, then h is a geodesic if and only if along h

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

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

ḣφφḣ+φḣφḣ=ḣḣ.E28

Since Us and Xs are the vertical and horizontal parts of the tangent vector field ḣs=W of 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 P and Q are 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 h is geodesic if and only if (26, 27) hold.

Theorem 1.4 [34] Let π be an anti-invariant submersion from a nearly Kähler manifold Mφg onto a Riemannian manifold Ngn. Also, let h:JRM be a regular curve and Us and Xs are the vertical and horizontal parts of the tangent vector field ḣs=W of hs. Then π is a Clairaut submersion with r=ef if and only if along h

ggradfXgUU=gHḣβX+AXαX+TUαX+PḣsUφU.

Proof: Let h:JRM be a geodesic on M and =ḣs2. Let θs be the angle between ḣs and 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, π is a Clairaut submersion with r=ef if 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 π be an Clairaut anti-invariant submersion from a nearly Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. Then

AφWφX+QWφX=XfW

for Xkerπ, Wkerπ and φW is basic.

Proof: Let π be an anti-invariant submersion from a nearly Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. We know that any fiber of Riemannian submersion π is totally umbilical if and only if

TVW=gVWH,E37

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

TVW=gVWgradf.E38

Let Xμ and 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 φW is basic, so HVφW=AφWV, therefore we have

gAφWVφX=gVWgradfX+gVQWφX,
gVAφWφX+gVQWφX=gVWgradfXE42

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

Theorem 1.6 [34] Let π be a Clairaut anti-invariant submersion from a nearly Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef and gradfφkerπ. Then either f is constant on φkerπ or the fibers of π are 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=U in (44), we have

gVVggradfφW=gVWggradfφV.E45

Interchange V with W in (45), we have

gWWggradfφV=gVWggradfφW.E46

By (45) and (46), we have

g2VWggradfφV=gVVgWWggradfφV.

Therefore either f is constant on φkerπ or V=aW, where a is constant (by using Schwarz’s Inequality for equality case).

Corollary 1.1 [34] Let π be a Clairaut anti-invariant submersion from a nearly Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef and gradfφkerπ. If dimkerπ>1, then the fibers of π are totally geodesic if and only if AφWφX+QWφX=0 for Wkerπ such that φW is basic and Xμ.

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

Corollary 1.2 [34] Let π be an Clairaut Lagrangian submersion from a nearly Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. Then either the fibers of π are 1-dimensional or they are totally geodesic.

Proof: Let π be an Clairaut Lagrangian submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. Then μ=0, so AφWφX+QWφX=0 always.

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

Example 1.2 [34] Let R4φg be a nearly Kähler manifold endowed with Euclidean metric g on R4 given 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 R3g1 be a Riemannian manifold endowed with metric g1=i=13dyi2.

  1. Consider a map π:R4φgR3g1 defined 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 π is an anti-invariant Riemannian submersion. Since the fibers of π are 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=0 for all X,YΓkerπ. Therefore fibers of π are totally geodesic. Thus π is Clairaut trivially.

  1.   ii. Consider a map π:R4φgR3g1 defined 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 π is an anti-invariant Riemannian submersion. Since the fibers of π are 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+x22 on R4φg, the gradient of f with respect to g is 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 π is an proper Clairaut anti-invariant submersion with r=ef for f=lnx12+x22.

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

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6. Semi-invariant Riemannian submersion

Definition 1.4 Let Mφg be an almost Hermitian manifold and N be a Riemannian manifold with Riemannian metric gn. A Riemannian submersion π:MN is called a semi-invariant Riemannian submersion [11] if there is a distribution D1kerπ such that

kerπ=D1D2andφD1=D1,φD2kerπ,

where D2 is orthogonal complementary to D1 in kerπ. For VΓkerπ, we have

φV=ϕV+ωV,E47

where ϕVΓD1 and ωVΓφD2.

Definition 1.5 A semi-invariant Riemannian submersion π is said to be a Lagrangian Riemannian submersion [4] if φkerπ=kerπ. Hence, if π is a Lagrangian Riemannian submersion then for any VΓkerπ, φV=ωV, ϕV=0 and 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 π be a semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn. If h:JRM is a regular curve and Us and Xs are the vertical and horizontal parts of the tangent vector field ḣs=W of hs, respectively, then h is 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 π be a semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn. Since φ2ḣ=ḣ. Taking the covariant derivative of this and using (2), we have

φḣφḣ=ḣḣ.E50

Since Us and Xs are the vertical and horizontal parts of the tangent vector field ḣs=W of 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 h is geodesic if and only if (48) and (49) hold.

Theorem 1.8 [40] Let π be a semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn. Also, let h:JRM be a regular curve. Us and Xs are the vertical and horizontal parts of the tangent vector field ḣs=W of hs. Then π is a Clairaut submersion with r=ef if and only if along h

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

Proof: Let h:JRM be a geodesic on M and =ḣs2. Let θs be the angle between ḣs and 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, π is a Clairaut submersion with r=ef if 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 π be a Clairaut semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. Then

gAωWVβX+gHVϕWβX+gVVωWαX+ĝVϕWαX=gVWXf

for XΓμ, V,WΓD2 and ωW is basic.

Proof: Let π be a Clairaut semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. We know that any fiber of Riemannian submersion π is totally umbilical if and only if

TVW=gVWH,E57

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

TVW=gVWgradf.E58

Let Xμ and 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 ωW is basic, so HVωW=AωWV, therefore we have

gAωWVβX+gHVϕWβX+gVVωWαX+ĝVϕWαX=gVWgradfX.E60

Theorem 1.10 [40] Let π be a Clairaut semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef and V,WΓD1 Then gradfΓφkerπ.

Proof: Let V,WΓD1 and 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 U and V in (61) and adding the resulting equation with (61), we get

gVUggradfφX=0,

which gives ggradfφX=0. Therefore gradfΓφkerπ.

Theorem 1.11 [40] Let π be a Clairaut semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef and gradfΓφkerπ. Then either f is constant on φkerπ or the fibers of π are 1-dimensional.

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

gVUφU=gVUggradfφU,

which gives

gφVUU=gVUggradfφU,

since kerπ is 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 V and U in (63), we have.

gVVggradfφU=gVUggradfφV.E64

By (63) and (64), we have

g2VUggradfφU=gVVgUUggradfφU.

Therefore either f is constant on φkerπ or V=aU, where a is constant (by using Schwarz’s Inequality for equality case).

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

Theorem 1.12 [40] Let π be a Clairaut semi-invariant submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. If dim D2>1, then fibers are totally geodesic.

Corollary 1.3 [40] Let π be a Clairaut Lagrangian submersion from a Kähler manifold Mφg onto a Riemannian manifold Ngn with r=ef. Then either the fibers of π are 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 R6φg be a Kähler manifold endowed with Euclidean metric g on R6 given 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 R3g1 be a Riemannian manifold endowed with metric g1=i=13dyi2.

Consider a map π:R6φgR3g1 defined 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=X2 therefore D1=spanX2X3 and D2=spanX1. Thus, we can say that π is 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=0 for all X,YΓkerπ. Therefore fibers of π are totally geodesic. Thus π is Clairaut trivially.

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Classification

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

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Written By

Sanjay Kumar Singh and Punam Gupta

Reviewed: 28 October 2021 Published: 24 December 2021