# A Study on Conservative C-Bochner Curvature Tensor in K-Contact and Kenmotsu Manifolds Admitting Semisymmetric Metric Connection

Authors: Gurupadavva Ingalahalli; C. S. Bagewadi
Journal: ISRN Geometry
 (2012)

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## Abstract

The paper deals with the study on conservative C-Bochner curvature tensor in K-contact and Kenmotsu manifolds admitting semisymmetric
metric connection, and it is shown that these manifolds areη-Einstein with respect to Levi-Civita connection, and the results are illustrated with examples.


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## Body

## 1. Introduction

In 1924, Friedmann and Schouten [1] introduced the idea of semisymmetric linear connection on a differentiable manifold. In 1932, Hayden [2] introduced the idea of metric connection with torsion on a Riemannian manifold. A systematic study of the semisymmetric metric connection on a Riemannian manifold was published by Yano [3] in 1970. After that the properties of semisymmetric metric connection have been studied by many authors like Amur and Pujar [4], Bagewadi [5], Sharfuddin and Hussain [6], De and Pathak [7], and so forth.A K-contact manifold is a differentiable manifold with a contact metric structure such thatξ is a Killing vector field ([8, 9]). These are studied by many authors like ([8–11]). The notion of Kenmotsu manifolds was defined by Kenmotsu [12]. Kenmotsu proved that a locally Kenmotsu manifold is a warped product I×fN of an interval I and a Kaehler manifold N with warping function f(t)=set, where s is a nonzero constant. For example it is hyperbolic space (-1). Kenmotsu manifolds were studied by many authors such as Binh et al. [13], Bagewadi and Venkatesha [14].In this paper we study conservative C-Bochner curvature tensor in K-contact and Kenmotsu manifolds admitting semisymmetric metric connection and obtain results on K-contact and Kenmotsu manifolds with respect to Levi-Civita connection.


## 2. Preliminaries

Ann-dimensional differential manifold M is said to have an almost contact structure (ϕ,ξ,η) if it carries a tensor field ϕ of type (1,1), a vector field ξ, and 1-form η on M, respectively, such that(2.1)ϕ2=-I+η⊗ξ,η(ξ)=1,η∘ϕ=0,ϕξ=0.Thus a manifoldM equipped with this structure is called an almost contact manifold [9] and is denoted by (M,ϕ,ξ,η). If g is a Riemannian metric on an almost contact manifold M such that,(2.2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),g(X,ξ)=η(X),
where X and Y are vector fields defined on M, then M is said to have an almost contact metric structure (ϕ,ξ,η,g) and M with this structure is called an almost contact metric manifold and is denoted by (M,ϕ,ξ,η,g).If on(M,ϕ,ξ,η,g) the exterior derivative of 1-form η satisfies,(2.3)dη(X,Y)=g(X,ϕY),
then (ϕ,ξ,η,g) is said to be a contact metric structure and M equipped with a contact metric structure is called contact metric manifold.If moreoverξ is Killing vector field, then M is called a K-contact manifold [8, 9]. A K-contact manifold is called Sasakian [9], if the relation(2.4)(∇Xϕ)Y=g(X,Y)ξ-η(Y)X
holds, where ∇ denotes the operator of covariant connection with respect to g.An almost contact metric manifold, which satisfies the following conditions,(2.5)(∇Xϕ)Y=-η(Y)ϕX-g(X,ϕY)ξ,∇Xξ=X-η(X)ξ,
where ∇ denotes the Riemannian connection of g hold, (M,ϕ,ξ,η,g) is called a Kenmotsu manifold.In a K-contact manifoldM, the following properties hold:(2.6)∇Xξ=-ϕX,(2.7)g(R(ξ,X)Y,ξ)=g(X,Y)-η(X)η(Y),(2.8)R(ξ,X)ξ=-X+η(X)ξ,(2.9)S(X,ξ)=(n-1)η(X),
where R is the Riemannian curvature tensor, S is the Ricci tensor and Q is the Ricci operator of M, respectively.In a Kenmotsu manifoldM, the following properties hold [12]:
(2.10)∇Xξ=X-η(X)ξ,(2.11)R(X,Y)ξ=η(X)Y-η(Y)X,(2.12)S(X,ξ)=-(n-1)η(X),(2.13)Qξ=-(n-1)ξ,
where R is the Riemannian curvature tensor, S is the Ricci tensor, and Q is the Ricci operator of M, respectively.LetM be an n-dimensional Riemannian manifold of class C∞ with metric tensor g and let ∇ be the Levi-Civita connection on M. A linear connection ∇̃ in an almost contact metric manifold M is said to be a semisymmetric connection if the torsion tensor T of the connection ∇̃ satisfies(2.14)T(X,Y)=π(Y)X-π(X)Y,
where π is a 1-form on M with ρ as associated vector field, that is, π(X)=g(X,ρ) for any differentiable vector field X on M.A semisymmetric connection∇̃ is called semisymmetric metric connection if it further satisfies ∇̃g=0.In an almost contact manifold semisymmetric metric connection is defined by identifying the 1-form of (2.14) with the contact-form η, that is,(2.15)T(X,Y)=η(Y)X-η(X)Y,
with ξ as associated vector field, that is, g(X,ξ)=η(X).The relation between the semisymmetric metric connection∇̃ and the Levi-Civita connection ∇ of M has been obtained by Yano [3], which is given by(2.16)∇̃XY=∇XY+η(Y)X-g(X,Y)ξ.
The above condition satisfies K-contact and Kenmotsu manifolds also.We denoteR, S, and r by curvature tensor, Ricci tensor, and scalar curvature with respect to Levi-Civita connection and correspondingly R̃, S̃, and r̃ with respect to semisymmetric metric connection. If B denotes C-Bochner curvature tensor [15] with respect to Levi-Civita connection B̃ with respect to semisymmetric metric connection is given by(2.17)B̃(X,Y)Z=R̃(X,Y)Z+1n+3×[g(X,Z)Q̃Y-S̃(Y,Z)X-g(Y,Z)Q̃X+S̃(X,Z)Y+g(ϕX,Z)Q̃ϕY-S̃(ϕY,Z)ϕX-g(ϕY,Z)Q̃ϕX+S̃(ϕX,Z)ϕY+2S̃(ϕX,Y)ϕZ+2g(ϕX,Y)Q̃ϕZ+η(Y)η(Z)Q̃X-η(Y)S̃(X,Z)ξ+η(X)S̃(Y,Z)ξ-η(X)η(Z)Q̃Y]-D̃+n-1n+3[g(ϕX,Z)ϕY-g(ϕY,Z)ϕX+2g(ϕX,Y)ϕZ]+D̃n+3[η(Y)g(X,Z)ξ-η(Y)η(Z)X+η(X)η(Z)Y-η(X)g(Y,Z)ξ]-D̃-4n+3[g(X,Z)Y-g(Y,Z)X],
where D̃=(n-1+r̃)/(n+1).Differentiate (2.17) covariantly with respect to ∇̃ and then contracting we get(2.18)(Div⋅B̃)=n+2n+3[(∇̃XS̃)(Y,Z)-(∇̃YS̃)(X,Z)]+1n+3×[(∇̃ϕYS̃)(ϕX,Z)-(∇̃ϕXS̃)(ϕY,Z)+2(∇̃ϕZS̃)(ϕX,Y)-(∇̃ξS̃)(X,Z)η(Y)-S̃(X,Z)(∇̃ξη)(Y)-(∇̃ξS̃)(Y,Z)η(X)+S̃(Y,Z)(∇̃ξη)(X)]+(n-1)2(n+1)(n+3)[(∇̃Yr̃)g(ϕX,ϕZ)-(∇̃Xr̃)g(ϕY,ϕZ)]+n(n+1)(n+3)[(∇̃ϕYr̃)g(ϕX,Z)-(∇̃ϕXr̃)g(ϕY,Z)+2(∇̃ϕZr̃)g(ϕX,Y)]-r̃+(n-1)(n+2)(n+1)(n+3)[η(∇̃ϕXξ)g(ϕY,Z)-η(∇̃ϕYξ)g(ϕX,Z)-2η(∇̃ϕZξ)g(ϕX,Y)]+r̃+(n-1)(n+1)(n+3)[(∇̃ξη)(Y)g(X,Z)-(∇̃ξη)(X)g(Y,Z)+(∇̃Yη)(Z)η(X)-(∇̃Xη)(Z)η(Y)+{(∇̃Yη)(X)-(∇̃Xη)(Y)}η(Z)]+(∇̃ξr̃)(n+1)(n+3)[g(X,Z)η(Y)-g(Y,Z)η(X)].


## 3. Relation betweenR, S, rand R̃, S̃, r̃ in a K-Contact Manifold

A relation between the curvature tensorR and R̃ of type (1,3) of the connections ∇ and ∇̃ by using (2.16) is given by(3.1)R̃(X,Y)Z=R(X,Y)Z+[g(ϕY,Z)X-g(ϕX,Z)Y]+[g(Y,Z)ϕX-g(X,Z)ϕY]+[g(ϕX,ϕZ)Y-g(ϕY,ϕZ)X]+[g(Y,Z)η(X)-g(X,Z)η(Y)]ξ.
From (3.1), it follows that(3.2)S̃(Y,Z)=S(Y,Z)+(n-2)g(ϕY,Z)-(n-2)g(Y,Z)+(n-2)η(Y)η(Z),
where S̃ denotes the Ricci tensor with respect to semisymmetric metric connection and S denotes the Ricci tensor. On contacting (3.2), we get(3.3)r̃=r-(n-1)(n-2),
where r̃ and r are scalar curvatures with respect to semisymmetric metric connection and Levi-Civita connection.In a Riemannian manifoldM, ξ is a Killing vector field in K-contact manifold, that is, S and r are invariant under it, that is,(3.4)(Lξg)(X,Y)=g(∇Xξ,Y)+g(X,∇Yξ)=0,(3.5)(LξS)(X,Y)=(Lξg)(QX,Y)=0,(3.6)(Lξr)=0,
where L is Lie-derivative. We know that(3.7)(∇ξS)(Y,Z)=ξS(Y,Z)-S(∇ξY,Z)-S(Y,∇ξZ)=(LξS)(Y,Z)-S(∇Yξ,Z)-S(Y,∇Zξ),
by using (3.5) in (3.7) and by virtue of (2.6), we have(3.8)(∇ξS)(Y,Z)=0.
Now in a K-contact manifold L̃, S̃, and r̃ are with respect to semisymmetric metric connection, that is,(3.9)(L̃ξg)(X,Y)=2[g(X,Y)-η(X)η(Y)]≠0,(L̃ξS̃)(X,Y)=2(n-2)g(ϕX,Y)+2g(X,Y)-2η(X)η(Y)≠0,(L̃ξr̃)≠0.
This shows that it is not K-contact with respect to a semisymmetric metric connection.Now from (3.2), we have(3.10)(∇̃WS̃)(Y,Z)=(∇WS)(Y,Z)+2(n-2)g(Y,W)η(Z)-2(n-2)η(Y)g(Z,ϕW)-2(n-2)η(Y)η(Z)η(W)-η(Y)S(W,Z)-η(Z)S(Y,W)+(n-1)g(W,Y)η(Z)+(n-1)g(W,Z)η(Y).
Put W=ξ in (3.10), we have(3.11)(∇̃ξS̃)(Y,Z)=(∇ξS)(Y,Z),(∇̃ξr̃)=(∇ξr).
But(3.12)(∇ξS)(Y,Z)=0⟹(∇̃ξS̃)(Y,Z)=0,(3.13)(∇ξr)=0⟹(∇̃ξr̃)=0.
Put Z=ξ in (3.10), we have(3.14)(∇̃WS̃)(Y,ξ)=S(Y,ϕW)-S(Y,W)+(n-1)g(Y,W)-(n-1)g(Y,ϕW)+2(n-2)g(ϕY,ϕW).


## 4. K-Contact Manifold Admitting Semisymmetric Metric Connection withDiv·B̃=0

ConsideringDiv·B̃=0 in (2.18) and putting X=ξ in the equation; using (3.2), (3.3), (3.12), (3.13), and (3.14) and by virtue of (2.1), (2.6), and (2.16), we get(4.1)n+2n+3[-S(ϕY,Z)+(3n-5)g(Z,ϕY)+S(Y,Z)-(n-1)g(Y,Z)]-(n-1)2(n+1)(n+3)[(∇ξr)g(ϕY,ϕZ)]+r-(n-1)(n-3)(n+1)(n+3)[g(Y,ϕZ)+g(Y,Z)-η(Y)η(Z)]+(∇ξr)(n+1)(n+3)[η(Y)η(Z)-g(Y,Z)]=0.
In a K-contact manifold, ξ[r]=0, that is, ∇ξr=0 then the above equation reduces to(4.2)n+2n+3[-S(ϕY,Z)+(3n-5)g(Z,ϕY)+S(Y,Z)-(n-1)g(Y,Z)]+r-(n-1)(n-3)(n+1)(n+3)[g(Y,ϕZ)+g(Y,Z)-η(Y)η(Z)]=0.
Interchanging Y and Z in the above equation then we have(4.3)n+2n+3[-S(ϕZ,Y)+(3n-5)g(Y,ϕZ)+S(Z,Y)-(n-1)g(Z,Y)]+r-(n-1)(n-3)(n+1)(n+3)[g(Z,ϕY)+g(Z,Y)-η(Y)η(Z)]=0.
Adding these equation (4.2) and (4.3), we have(4.4)S(Y,Z)=[(n-1)(n2+4n-1)-r(n+1)(n+2)]g(Y,Z)+[r-(n-1)(n-3)(n+1)(n+2)]η(Y)η(Z).
Then the above equation is written as S(Y,Z)=αg(Y,Z)+βη(Y)η(Z), where(4.5)α=[(n-1)(n2+4n-1)-r(n+1)(n+2)],β=[r-(n-1)(n-3)(n+1)(n+2)].
On contracting (4.4), we get(4.6)R1=(n-1)[n(n2+4n-1)-r-(n-3)](n+1)(n+2).
Hence we state the following theorem.Theorem 4.1.
If in a K-contact manifold the C-Bochner curvature tensor with respect to semisymmetric metric connection is conservative, then the manifold isη-Einstein with respect to Levi-Civita connection and the scalar curvature of such a manifold is given in (4.6).


## 5. Example for K-Contact Manifold

Consider the 3-dimensional manifoldM={(x,y,z):(x,y,z)∈R3}, where (x,y,z) are the standard coordinates in R3. Let (E1,E2,E3) be linearly independent at each point of M(5.1)E1=∂∂x+y∂∂z,E2=∂∂y-x∂∂z,E3=∂∂z.
Let g be the Riemannian metric defined by(5.2)g(E1,E2)=g(E2,E3)=g(E1,E3)=0,g(E1,E1)=g(E2,E2)=g(E3,E3)=1,
where g is given by g=[(1-y2)dx⊗dx+(1-x2)dy⊗dy+dz⊗dz]. Let ξ be the vector field, η be the 1-form, and ϕ be the (1,1) tensor field defined by(5.3)ξ=∂∂z,η=dz+xdy-ydx,ϕE1=-E2,ϕE2=E1,ϕE3=0.
The linearity property of ϕ and g yields that(5.4)η(E3)=1,ϕ2U=-U+η(U)E3,g(ϕU,ϕW)=g(U,W)-η(U)η(W),
for any vector fields U, W on M. Thus for E3=ξ. The structure (ϕ,ξ,η,g) defines on M. By definition of Lie bracket, we have(5.5)[E1,E2]=-2E3,[E1,E3]=[E2,E3]=0.
Let ∇ be Levi-Civita connection with respect to the above metric g given by Koszul formula, that is(5.6)2g(∇XY,Z)=X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))-g(X,[Y,Z])-g(Y,[X,Z])+g(Z,[X,Y]).
Then by Koszula formula, we have(5.7)∇E1E1=0,∇E2E2=0,∇E3E3=0,∇E1E2=-E3,∇E2E1=-E3,∇E2E3=-E1,∇E1E3=E2,∇E3E1=E2,∇E3E2=-E1.
Clearly one can see that (ϕ,ξ,η,g) is a K-contact structure.The Ricci tensorS(X,Y) is given by(5.8)S(X,Y)=∑i=13g(R(X,Ei)Ei,Y)=g(R(X,E1)E1,Y)+g(R(X,E2)E2,Y)+g(R(X,E3)E3,Y).
The nonzero components of R(X,Ei)Ei, where i=1, 2, 3, and by virtue of (5.7) we have(5.9)R(E2,E1)E1=-3E2,R(E3,E1)E1=E3,R(E1,E2)E2=-3E1,R(E3,E2)E2=E3,R(E1,E3)E3=E1,R(E2,E3)E3=E2.
Using these in (5.8), we have(5.10)S(X,Y)=-2g(X,Y)+4η(X)η(Y).
This shows that R3 is an η-Einstein. This is an example of K-contact manifold which is an η-Einstein.IfX=Y=Z=Ei, in (2.18) and by virtue of (3.2), we obtain (Div·B̃)=0. Thus Theorem 4.1 holds true.However, ifX≠Y≠Z=Ei, in (2.18) and by virtue of (3.2), we obtain (Div·B̃)≠0. Hence in general, if X=∑i=13aiEi,  Y=∑i=13biEi and Z=∑i=13ciEi, then (Div·B̃)(X,Y)Z≠0. In this case the converse of Theorem 4.1 does not hold true.


## 6. Relation betweenR, S, randR̃, S̃, r̃ in a Kenmotsu Manifold

A relation between the curvature tensorR and R̃ of type (1,3) of the connections ∇ and ∇̃ by using (2.16) is given by(6.1)R̃(X,Y)Z=R(X,Y)Z+[g(X,Z)Y-g(Y,Z)X]+2[g(ϕX,ϕZ)Y-g(ϕY,ϕZ)X]+2[g(Y,Z)η(X)-g(X,Z)η(Y)]ξ.
From (6.1), it follows that(6.2)S̃(Y,Z)=S(Y,Z)-(n-1)g(Y,Z)-2(n-2)g(ϕY,ϕZ),
where S̃ denotes the Ricci tensor with respect to semisymmetric metric connection and S denotes the Ricci tensor. On contacting (6.2), we get(6.3)r̃=r-(n-1)(3n-4),
where r̃ and r are scalar curvatures with respect to semisymmetric metric connection and Levi-Civita connection.In a Kenmotsu manifoldM, ξ is a unit vector field in Kenmotsu manifold, then the following properties hold(6.4)(Lξg)(X,Y)=2[g(X,Y)-η(X)η(Y)],(LξS)(X,Y)=2[S(X,Y)+(n-1)η(X)η(Y)],(Lξr)=(Divξ)r,
where L is a Lie derivative. We know that(6.5)(∇ξS)(Y,Z)=ξS(Y,Z)-S(∇ξY,Z)-S(Y,∇ξZ)=(LξS)(Y,Z)-S(∇Yξ,Z)-S(Y,∇Zξ).
By using (6.4) in (6.5) and by virtue of (2.10) we have
(6.6)(∇ξS)(Y,Z)=0,(6.7)(∇ξr)=0.
Now in a Kenmotsu manifold L̃, S̃, and r̃ are with respect to semisymmetric metric connection, then we define the properties like(6.8)(L̃ξg)(X,Y)=4[g(X,Y)-η(X)η(Y)],(L̃ξS̃)(X,Y)=4[S(X,Y)-(n-1)g(ϕX,ϕY)-2(n-2)g(ϕX,ϕY)+(n-1)η(X)η(Y)],L̃ξr̃=(Divξ̃)r̃.
Now from (6.2), we have(6.9)(∇̃XS̃)(Y,Z)=(∇XS)(Y,Z)-(n-1)[g(X,Y)η(Z)+g(X,Z)η(Y)]-[η(Y)S(X,Z)+η(Z)S(X,Y)]+4(n-2)×[η(Y)g(ϕX,ϕZ)+g(ϕX,ϕY)η(Z)].
Putting X=ξ in the above equation and by virtue of (6.6) then we have
(6.10)(∇̃ξS̃)(Y,Z)=0,(6.11)(∇̃ξr̃)=0.
Again putting Z=ξ in (6.9), we have(6.12)(∇̃XS̃)(Y,ξ)=-2S(Y,X)-2(n-1)g(Y,X)+4(n-2)g(ϕY,ϕX).


## 7. Kenmotsu Manifold Admitting Semisymmetric Metric Connection withDiv·B̃=0

ConsideringDiv·B̃=0 in (2.18) and putting X=ξ in the equation; using (6.2), (6.3), (6.10), and (6.12) and by virtue of (2.10), (2.16) and on simplification we get(7.1)n+2n+3[2S(Y,Z)+2(n-1)g(Y,Z)-4(n-2)g(ϕY,ϕZ)]-(n-1)2(n+1)(n+3)[(∇ξr)g(ϕY,ϕZ)]+r-(n-1)(3n-4)+(n-1)(n+1)(n+3)×[2g(Y,Z)-2η(Y)η(Z)]-(∇ξr)(n+1)(n+3)[g(ϕY,ϕZ)]=0.
In a Kenmotsu manifold M, ∇ξr=0 then the above equation reduces to(7.2)n+2n+3[2S(Y,Z)+2(n-1)g(Y,Z)-4(n-2)g(ϕY,ϕZ)]+r-(n-1)(3n-5)(n+1)(n+3)[2g(Y,Z)-2η(Y)η(Z)]=0.
On simplifying the above equation we get(7.3)S(Y,Z)=[(n-3)-r-(n-1)(3n-5)(n+1)(n+2)]g(Y,Z)+[-2(n-2)+r-(n-1)(3n-5)(n+1)(n+2)]η(Y)η(Z).
Then the above equation is written as S(Y,Z)=αg(Y,Z)+βη(Y)η(Z), where(7.4)α=[(n-3)-r-(n-1)(3n-5)(n+1)(n+2)],β=[-2(n-2)+r-(n-1)(3n-5)(n+1)(n+2)].
On contracting (7.3), we get(7.5)r1=[(n-4)(n-1)-(n-1)[r-(n-1)(3n-5)](n+1)(n+2)].
Hence we state the following theorem.Theorem 7.1.
If in a Kenmotsu manifold the C-Bochner curvature tensor with respect to semisymmetric metric connection is conservative, then the manifold isη-Einstein with respect to Levi-Civita connection, and the scalar curvature of such a manifold is given in (7.5).


## 8. Example for Kenmotsu  Manifold

LetM={(x,y,z)∈R3}. Let (E1,E2,E3) be linearly independent vector fields given by(8.1)E1=e-z(∂∂x+∂∂y),E2=e-z(-∂∂x+∂∂y),E3=∂∂z.
Let g be the Riemannian metric defined by(8.2)g(E1,E2)=g(E2,E3)=g(E1,E3)=0,g(E1,E1)=g(E2,E2)=g(E3,E3)=1,
where g is given by g=(e2z/2)(dx⊗dx+dy⊗dy)+dz⊗dz. Let ξ be the vector field, η be the 1-form, and ϕ be the (1,1) tensor field defined by(8.3)η=dz,ξ=E3=∂∂z,ϕE1=E2,ϕE2=-E1,ϕE3=0.
The linearity property of ϕ and g yields that(8.4)η(E3)=1,ϕ2U=-U+η(U)E3,g(ϕU,ϕW)=g(U,W)-η(U)η(W),
for any vector fields U, W on M. By definition of Lie bracket, we have(8.5)[E1,E2]=0,[E2,E3]=E2,[E1,E3]=E1.
Let ∇ be the Levi-Civita connection with respect to above metric g which is given by Koszula formula (5.6), and by virtue of it we have(8.6)∇E1E3=E1,∇E2E3=E2,∇E3E3=0,∇E1E2=0,∇E2E2=-E3,∇E3E2=0,∇E1E1=-E3,∇E2E1=0,∇E3E1=0.
Clearly one can see that (ϕ,ξ,η,g) is a Kenmotsu structure.The nonzero components ofR(X,Ei)Ei, where i=1, 2, 3, and by virtue of (8.6) we have(8.7)R(E1,E2)E2=-E1,R(E1,E3)E3=-E1,R(E2,E1)E1=-E2,R(E2,E3)E3=-E2,R(E3,E1)E1=-E3,R(E3,E2)E2=-E3.
The Ricci tensor S(X,Y) is given in (5.8) by virtue of (8.7), and we have(8.8)S(X,Y)=-2g(X,Y)+0⋅η(X)η(Y).
This shows that R3 is an η-Einstein. This is an example of Kenmotsu manifold which is an η-Einstein.IfX=Y=Z=Ei, in (2.18) and by virtue of (6.2), we obtain (Div·B̃)=0. Thus Theorem 7.1 holds true.However, ifX≠Y≠Z=Ei, in (2.18) and by virtue of (6.2), we obtain (Div·B̃)≠0. Hence in general, if X=∑i=13aiEi, Y=∑i=13biEi and Z=∑i=13ciEi, then (Div·B̃)(X,Y)Z≠0. In this case the converse of Theorem 7.1 does not hold true.



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*Source: 709243-2012-03-05.xml*