# D

Authors: H. G. Nagaraja; C. R. Premalatha
Journal: ISRN Geometry
 (2013)

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## Abstract

We studyDa-homothetic deformations of K-contact manifolds. We prove that Da-homothetically deformed K-contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature of Da-homothetically deformed K-contact manifolds.


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## Body

## 1. Introduction

In 1968 Tanno [1] introduced the notion of Da-homothetic deformations. Carriazo and Martín-Molina [2] studied Da-homothetic deformation of generalized (k,μ) space forms and gave several examples for manifolds of dimension 3. De and Ghosh [3] studied Da-homothetic deformation of almost normal contact metric manifolds and prove that Qϕ-ϕQ is invariant under such transformation. Bagewadi and Venkatesha [4] studied concircularly semisymmetric trans-Sasakian manifolds and De et al. [5] studied conharmonically semisymmetric, conharmonically flat, ξ-conharmonically flat, and conharmonically recurrent generalized Sasakian space forms. Several authors [6–11] studied K-contact manifolds and proved conditions for these manifolds to be of ξ-conformally flat, ϕ-conformally flat, quasi-conharmonically flat, and ξ-conharmonically flat. Motivated by the above studies, in this paper we study Da-homothetic deformations of K-contact manifolds by considering conharmonic and projective curvature tensor. The paper is organized as follows. After Preliminaries, we give a brief account of information of Da-homothetic deformation of K-contact manifolds in Section 3. In Section 4, we study conharmonically flat, semisymmetric, ϕ-conharmonically flat, quasi-conharmonically flat, and ξ-conharmonically flat K-contact manifolds with respect to Da-homothetic deformation. In the last section, we consider Weyl projective curvature in K-contact manifolds with respect to Da-homothetic deformation.


## 2. Preliminaries

Let(M,ϕ,ξ,η,g) be a (2n+1)-dimensional almost contact metric manifold [12], consisting of a (1,1) tensor field ϕ, a vector field ξ, a 1-form η, and Riemannian metric g. Then
(1)ϕ2=-I+η⊗ξ,η(ξ)=1,ϕξ=0,η∘ϕ=0,(2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),(3)g(X,ϕY)=-g(ϕX,Y),g(X,ϕX)=0,g(X,ξ)=η(X),
for all X, Y∈TM. If ξ is a Killing vector field, then M is called a K-contact Riemannian manifold [13]. A K-contact Riemannian manifold is called Sasakian [12], if the relation
(4)(∇Xϕ)Y=g(X,Y)ξ-η(Y)X
holds, where ∇ denotes the operator of covariant differentiation with respect to g.IfM2n+1 is a K-contact Riemannian manifold, then besides (1), (2), (3), and (4) the following relations hold [14]:
(5)∇Xξ=-ϕX,(6)(∇Xη)(Y)=-g(ϕX,Y),(7)S(X,ξ)=g(QX,ξ)=2nη(X),(8)η(R(X,Y)Z)=g(Y,Z)η(X)-g(X,Z)η(Y),(9)R(X,Y)ξ=η(Y)X-η(X)Y,(10)R(ξ,X)Y=g(X,Y)ξ-η(Y)X,
for any vector fields X and Y, where R and S denote, respectively, the curvature tensor of type (1,3) and the Ricci tensor of type (0,2).Definition 1.
A contact metric manifoldM is said to be η-Einstein if S(X,Y)=αg(X,Y)+βη(X)η(Y), where α and β are smooth functions on M.


## 3.Da-Homothetic Deformation of K-Contact Manifolds

Let(M,ϕ,ξ,η,g) be a (2n+1)-dimensional almost contact metric manifold. A Da-homothetic deformation is defined by
(11)ϕ-=ϕ,ξ-=1aξ,η-=aη,g-=ag+a(a-1)η⊗η,
with a being a positive constant [1].It is clear that the(M-,ϕ-,ξ-,η-,g-) is also an almost contact metric manifold.If(M,ϕ,ξ,η,g) is a K-contact manifold with Riemannian connection ∇, the connection ∇- of the Da-deformed K-contact manifold (M-,ϕ-,ξ-,η-,g-) can be calculated from ∇ and g-. Using Koszul's formula and (5), (6), and (11), ∇- of g- is given by
(12)∇-XY=∇XY-a(a-1)[η(Y)ϕX+η(X)ϕY].
Using (12), we obtain
(13)(∇-Xϕ)Y=(∇Xϕ)Y+(a-1)η(Y)ϕ2X.
The curvature tensor R- of (M,ϕ-,ξ-,η-,g-) is given by
(14)R-(X,Y)Z=R(X,Y)Z-(a-1)ddd×(g(ϕY,Z)ϕX+g(ϕZ,X)ϕY+2g(ϕY,X)ϕZdddddd+[g(X,Z)ξ-η(Z)X]η(Y)dddddd-[g(Y,Z)ξ-η(Z)Y]η(X)ddddsd.+a[η(Y)X-η(X)Y]η(Z)).
Using (9), (10), and (14), we have
(15)R-(X,Y)ξ-=(2-a)[η(Y)X-η(X)Y],R-(ξ-,Y)Z=[g(Y,Z)ξ-η(Z)Y]-(a-1)[η(Y)ξ-Y]η(Z),R-(ξ-,Y)ξ-=(2-a)a[η(Y)ξ-Y],η-(R-(X,Y)Z)=a2[g(Y,Z)η(X)-g(X,Z)η(Y)].
From (14), we get
(16)S-(Y,Z)=aS(Y,Z)-a(a-1)×((2-a)g(Y,Z)+[2n(a-1)+a-2]η(Y)η(Z)),
where S- and S are the Ricci tensors of (M-,ϕ-,ξ-,η-,g-) and (M,ϕ,ξ,η,g), respectively.It follows from (16) that
(17)S-(Y,ξ-)=2na(2-a)η(Y),(18)S-(ϕY,ϕZ)=S-(Y,Z)-2na2(2-a)η(Y)η(Z).
Again contracting (16) over Y, Z, we get
(19)r-=ar-2na(a-1),
where r- and r are the scalar curvatures of (M-,ϕ-,ξ-,η-,g-) and (M,ϕ,ξ,η,g), respectively.


## 4. Conharmonic Curvature Tensor inDa-Homothetically Deformed K-Contact Manifolds

The conharmonic tensor of aDa-homothetically deformed K-contact manifold is defined by [15]
(20)K-(X,Y)Z=R-(X,Y)Z-12n-1×[S-(Y,Z)X-S-(X,Z)Y+g-(Y,Z)Q-Xdd-g-(X,Z)Q-Y],
for X, Y, Z∈TM, where R-, S-, and Q- are the Riemannian curvature tensor, Ricci tensor, and Ricci operator of (M-,ϕ-,ξ-,η-,g-).Definition 2.
An almost contact metric manifold(M,ϕ,ξ,η,g) is said to be(1)
conharmonically flat if(21)K(X,Y)Z=0,(2)
conharmonically semisymmetric if(22)R·K=0,(3)
ϕ-conharmonically flat if
(23)g(K(ϕX,ϕY)ϕZ,ϕW)=0,(4)
quasi-conharmonically flat if(24)g(K(X,Y)Z,ϕW)=0,(5)
ξ-conharmonically flat if
(25)K(X,Y)ξ=0,
for all vector fields X, Y, and Z.Assume thatM- is conharmonically flat K-contact manifold with respect to Da-homothetic deformation. So, we have K-(X,Y)Z=0.Then from (20), we have
(26)R-(X,Y)Z=12n-1×[S-(Y,Z)X-S-(X,Z)Y+g-(Y,Z)Q-Xdd-g-(X,Z)Q-Y].
Setting Z=ξ-, contracting  (26) with W, and using (7), (9), (14), and (16), we obtain
(27)(2-a)(2n-1-2na)[η(Y)g-(X,W)-η(X)g-(Y,W)]=η-(Y)S-(X,W)-η-(X)S-(Y,W).
Taking Y=ξ- in (27) and using (1), (7), and (16), it follows that
(28)S-(X,W)=(2-a)(2n-1-2na)ag-(X,W)+(2-a)(4na-2n+1)aη-(X)η-(W).
Thus, M- is η-Einstein.Using (28) in (26), we obtain
(29)R-(X,Y,Z,W)=2(2-a)(2n-1-2na)a(2n-1)×[g-(Y,Z)g-(X,W)-g-(X,Z)g-(Y,W)]+(2-a)(4na-2n+1)a(2n-1)×([g-(X,W)η-(Y)-g-(Y,W)η-(X)]η-(Z)fddddd.+[g-(Y,Z)η-(X)-g-(X,Z)η-(Y)]η-(W)).
From (29), we get
(30)R-(X,Y)Z=2(2-a)(2n-1-2na)a(2n-1)×[g-(Y,Z)X-g-(X,Z)Y]-(2-a)(4na-2n+1)a(2n-1)×[η-(X)η-(Z)Y-η-(Y)η-(Z)Xds+g-(X,Z)η-(Y)ξ-g-(Y,Z)η-(X)ξ].
Hence, it reduces to a generalized Sasakian space form with f1=2(2-a)(2n-1-2na)/a(2n-1), f2=0, and f3=-(2-a)(4na-2n+1)/a(2n-1). Thus, (30) leads to the following.Theorem 3.
A conharmonically flatK-contact manifold admitting Da-homothetic deformation reduces to a generalized Sasakian space form with associated functions f1=2(2-a)(2n-1-2na)/a(2n-1), f2=0, and f3=-(2-a)(4na-2n+1)/a(2n-1).Let us now consider a conharmonically semisymmetricK-contact manifold admitting Da-homothetic deformation. Then the condition
(31)R-(X,Y)·K-=0
holds on M- for all vector fields X, Y.From (8), (14), (17), and (20), we obtain
(32)η-(K-(X,Y)Z)=a2[g(Y,Z)η(X)-g(X,Z)η(Y)]-1(2n-1)×[(S-(Y,Z)η-(X)-S-(X,Z)η-(Y))ddd.ddS-+2na(2-a)(g-(Y,Z)η(X)-g-(X,Z)η(Y))].
Setting Z=ξ-, in (32), we get
(33)η-(K-(X,Y)ξ-)=0.
Again taking X=ξ- in (32) and using (17), we obtain
(34)η-(K-(ξ-,Y)Z)=-12n-1S-(Y,Z)+[1-2n(2-a)2n-1]g-(Y,Z)+[-1+4n(2-a)2n-1]η-(Z)η-(Y).
Now, (21) yields
(35)R-(X,Y)K-(U,V)Z-K-(R-(X,Y)U,V)Z-K-(U,R-(X,Y)V)Z-K-(U,V)R-(X,Y)Z=0.
Therefore,
(36)g-(R-(ξ-,Y)K-(U,V)Z,ξ-)-g-(K-(R-(ξ-,Y)U,V)Z,ξ-)-g-(K-(U,R-(ξ-,Y)V)Z,ξ-)-g-(K-(U,V)R-(ξ-,Y)Z,ξ-)=0.
From this it follows that
(37)-K-(U,V,Z,Y)+η(K-(U,V)Z)η(Y)+[g(Y,U)-(a-1)η(U)η(Y)]η(K-(ξ,V)Z)+(a-2)η(U)η(K-(Y,V)Z)+[g(Y,V)-(a-1)η(V)η(Y)]η(K-(U,ξ)Z)+(a-2)η(V)η(K-(U,Y)Z)+[g(Y,Z)-(a-1)η(Z)η(Y)]η(K-(U,V)ξ)+(a-2)η(Z)η(K-(U,V)Y)=0,
where
(38)K-(U,V,Z,Y)=g-(K-(U,V)Z,Y).
Taking Y=U in (37) and making use of (32) and (33), we obtain
(39)-K-(U,V,Z,U)+(a-1)η(U)η(K-(U,V)Z)+[g(U,U)-(a-1)η(U)η(U)]η(K-(ξ,V)Z)+[g(U,V)-(a-1)η(U)η(V)]η(K-(U,ξ)Z)+(a-2)η(Z)η(K-(U,V)U)=0.
If {e1,e2,…,e2n,ξ-} is a local orthonormal basis of vector fields in M-, then, from (39), we get
(40)∑i=12n‍K-(ei,V,Z,ei)=(a-1)∑i=12n‍η(ei)η(K-(ei,V)Z)+∑i=12n‍[g(ei,ei)-(a-1)η(ei)η(ei)]η(K-(ξ,V)Z)+∑i=12n‍[g(ei,V)-(a-1)η(ei)η(V)]η(K-(ei,ξ)Z)+(a-2)∑i=12n‍η(K-(ei,V)ei)η(Z).
From (20), it follows that
(41)∑i=12n‍K-(ei,V,Z,ei)=12n-1S-(V,Z)-[1-r-+2n(2-a)2n-1]g-(V,Z)+[1-4n(2-a)2n-1]η-(V)η-(Z),∑i=12n‍η(ei)η(K-(ei,V)Z)=1-a2a2(2n-1)S-(V,Z)+[a2-1a2+2n(2-a)(1-a2)a2(2n-1)]g-(V,Z)+1-a2a2η-(V)η-(Z),∑i=12n‍[g(ei,V)-(a-1)η(ei)η(V)]η(K-(ei,ξ)Z)=12n-1S-(V,Z)+[-a2+2na2(2-a)2n-1]g(V,Z)+[a2-2na(2-a)(a+1)(a2-a+1)2n-1]η(V)η(Z),∑i=12n‍η(K-(ei,V)Z)η(Z)=[r-2n-1+4n(1-a)]η(V)η(Z).
Using (41) in (40), we obtain
(42)(2n+2-a)η(K-(ξ-,V)Z)=(a-1)(a2-1)a2(2n-1)S-(V,Z)+[r-2n-1+2na2-a(1+4n)+1+2na2(2n-1)]g-(V,Z)+[(a-2)r-a2(2n-1)+pa2(2n-1)]η-(V)η-(Z),
where
(43)p=-2na5+2na4+10na3+a2(8n2-18n)+a(14n-24n2+1)+16n2-6n-1.
In view of (34), (42) yields
(44)S-(V,Z)=-1a2(2n+1)+1-a[αg-(V,Z)+βη-(V)η-(Z)],
where
(45)α=r-a2+2na4+a3(-4n2-6n-1)+a2(4n2+8n+2)+a(-1-4n)+1+2n,β=(a-2)r--2na5-2na4+a3(8n2+24n+1)+a2(-4n2-32n-2)+a(14n-24n2+1)+16n2-6n-1.
Thus, M- is η-Einstein.If{e1,e2,…,e2n,ξ-} is a local orthonormal basis of vector fields in M-, then, from (44), we get
(46)r-=(-2na5+4n2a4+a3l+a2mcd+a(-324n2+8n)+20n2+2n)×(1-2a2(2n+1))-1,
where
(47)l=-8n3-8n2+16n,m=8n3+16n2-20n.
So, we can state the following.Theorem 4.
In a(2n+1)-dimensional conharmonically semisymmetric K-contact manifold admitting Da-homothetic deformation, scalar curvature r- is given by (46).Analogous to the definition ofϕ-conharmonically flat K-contact manifolds [8], we define ϕ-conharmonically flat K-contact manifolds with respect to Da-homothetic deformation. Let us assume that M is a ϕ-conharmonically flat K-contact manifold with respect to Da-homothetic deformation. It can be easily seen that
(48)g-(K-(ϕX,ϕY)ϕZ,ϕW)=0,
where X, Y, Z, W∈TM-.Using (20), (48) yields
(49)g-(R-(ϕX,ϕY)ϕZ,ϕW)=12n-1(S-(ϕY,ϕZ)g-(ϕX,ϕW)dddddddddd-S-(ϕX,ϕZ)g-(ϕY,ϕW)dddddddddd+S-(ϕX,ϕW)g-(ϕY,ϕZ)dddddddddd-S-(ϕY,ϕW)g-(ϕX,ϕZ)),
for all X, Y, Z, W∈TM-.If{e1,e2,…,e2n,ξ-} is a local orthonormal basis of vector fields in M-, then {ϕe1,ϕe2,…,ϕe2n,ξ-} is also a local orthonormal basis. So, using (1), (6), (14), (16), and (18), it can be easily verified that
(50)∑i=12n‍R-(ϕei,ϕY,ϕZ,ϕei)=S-(Y,Z)-ag(Y,Z)+[a-2na(2-a)]η(Y)η(Z),∑i=12n‍g-(ϕei,ϕei)=2n,∑i=12n‍S-(ϕei,ϕei)=r--2n(2-a),∑i=12n‍S-(ϕY,ϕei)g-(ϕei,ϕZ)=S-(ϕY,ϕZ).
For a local orthonormal basis {ϕe1,ϕe2,…,ϕe2n,ξ-} of vector fields in M-, putting X=W=ei in (49) and summing up with respect to i=1,2,…,2n+1, we have
(51)∑i=12n‍g-(R-(ϕei,ϕY)ϕZ,ϕei)=12n-1∑i=12n‍(S-(ϕY,ϕZ)g-(ϕei,ϕei)ddddddddd.ddd-S-(ϕei,ϕZ)g-(ϕY,ϕei)ddddddddd.ddd+S-(ϕei,ϕei)g-(ϕY,ϕZ)ddddddddd.ddd-S-(ϕY,ϕei)g-(ϕei,ϕZ)),
for all Y, Z∈TM-. The previous equation, in view of (50), becomes
(52)S-(ϕY,ϕZ)=[a(2n-1)+a(r--2n(2-a))]×[g(Y,Z)-η(Y)η(Z)],
for all Y, Z∈TM-.Using (2) and (18), (52) reduces to
(53)S-(Y,Z)=[r-+2n-1-2n(2-a)]g-(Y,Z)+[2n(2-a)(a2+2a-1)d.d-(2a-1)(r-+2n-1)(a2+2a-1)]η-(Y)η-(Z).
Setting Y=Z=ei in (53), summing up with respect to i=1,2,…,2n+1, and using (19), we obtain
(54)r-=-2na3+a(4n2+8n+2)-4n2-6n-22a-2n-1.
Replacing X by ϕX and Y by ϕY in (53) and using (54), we obtain S-(ϕX,ϕY)=(r-+2n-1-2n(2-a))g-(ϕX,ϕY) for all X,Y∈TM-.Now using the previous expression in (49), we obtain
(55)R-(ϕX,ϕY,ϕZ,ϕW)=2(r-+2n-1-2n(2-a))2n-1×(g-(ϕY,ϕZ)g-(ϕX,ϕW)-g-(ϕX,ϕZ)g-(ϕY,ϕW)),
for all X, Y, Z, W∈TM-.The converse is obvious. Thus we have the following.Theorem 5.
A(2n+1)-dimensional K-contact manifold is ϕ-conharmonically flat with respect to Da-homothetic deformation if and only if M- satisfies (55).From (20), we obtain
(56)g-(K-(X,Y)Z,ϕW)=R-(X,Y,Z,ϕW)-12n-1×(S-(Y,Z)g-(X,ϕW)-S-(X,Z)g-(Y,ϕW)dddddd+S-(X,ϕW)g-(Y,Z)-S-(Y,ϕW)g-(X,ϕZ)),
for all X, Y, Z, W∈TM-.Suppose thatM is quasi-conharmonically flat K-contact manifold with respect to Da-homothetic deformation; that is,
(57)g-(K-(X,Y)Z,ϕW)=0.
Then (56) reduces to
(58)R-(X,Y,Z,ϕW)=12n-1(S-(Y,Z)g-(X,ϕW)-S-(X,Z)g-(Y,ϕW)dddddddddd+S-(X,ϕW)g-(Y,Z)-S-(Y,ϕW)g-(X,Z)).
For a local orthonormal basis {e1,e2,…,e2n,ξ-} of vector fields in M-, putting X=ϕei and W=ei in (58) and summing up with respect to i=1,2,…,2n+1, we obtain
(59)∑i=12n‍R-(ϕei,Y,Z,ϕei)=12n-1∑i=12n‍(S-(Y,Z)g-(ϕei,ϕei)ddddddddddd-S-(ϕei,Z)g-(Y,ϕei)ddddddddddd+S-(ϕei,ϕei)g-(Y,Z)ddddddddddd-S-(Y,ϕei)g-(ϕei,Z)).
Using (2), (10), (14), and (17) in (59), we obtain
(60)S-(Y,Z)=[r-+2n-1-2n(2-a)]g-(Y,Z)-(2n-1)η-(Y)η-(Z).
Taking Z=ξ- and using (1) and (17), we obtain
(61)r-=2n(2-a)(a+1),
and using (61) in (19), we obtain
(62)r=4na.
Hence, we can state the following.Theorem 6.
LetM be a K-contact manifold. Suppose that M- is obtained from M by Da-homothetic deformation. If M- is quasi-conharmonically flat, then the scalar curvatures r- and r of M- and M are, respectively, given by (61) and (62).Suppose thatM- is ξ-conharmonically flat. Then from (20), we have
(63)R-(X,Y)ξ-=12n-1×([Q-Xη-(Y)-Q-Yη-(X)]2na(2-a)[η(Y)X-η(X)Y]dcc+[Q-Xη-(Y)-Q-Yη-(X)]).
Contracting the above equation with respect to W, we obtain
(64)R-(X,Y,ξ-,W)=12n-1([Q-Xη-(Y)-Q-Yη-(X)]2na(2-a)[η(Y)g-(X,W)-η(X)g-(Y,W)]dddddcdddd+[S-(X,W)η-(Y)-S-(Y,W)η-(X)]),
for all X, Y, Z, W∈TM-.For a local orthonormal basis{e1,e2,…,e2n,ξ-} of vector fields in M-, using (64), we obtain
(65)∑i=12n‍R-(ei,Y,ξ-,ei)=12n-1∑i=12n‍([S-(ei,ei)η-(Y)-S-(Y,ei)η-(ei)]2na(2-a)[η(Y)g-(ei,ei)-η(ei)g-(Y,ei)]dddddddddddd+[S-(ei,ei)η-(Y)-S-(Y,ei)η-(ei)]).
Therefore,
(66)2nS-(Y,ξ-)=[4n2(2-a)+r-]η-(Y).
Using (17) in (66), we obtain
(67)r-=0.
Taking Y=ξ- in (64) and using (10), (14), and (17), we obtain
(68)S-(X,W)=[2na-4n-1]g-(X,W)+[6n-4na+1]η-(X)η-(W).
From this we can conclude that M is η-Einstein. Thus, we have the following.Theorem 7.
ADa-Homothetically deformed ξ-conharmonically flat K-contact manifold is η-Einstein and its scalar curvature vanishes.


## 5.Da-Homothetic Deformation of ξ-Weyl Projectively Flat K-Contact Manifolds

Suppose that, in a(2n+1)-dimensional K-contact manifold M with Da-homothetic deformation, the Ricci tensor vanishes; that is,
(69)S-(X,Y)=0.
Then from (16), we have
(70)S(Y,Z)=(a-1)([2n(a-1)+1+a]η(Y)η(Z)3g(ϕY,ϕZ)-(1+a)g(Y,Z)dddddd+[2n(a-1)+1+a]η(Y)η(Z)).
The Weyl projective curvature tensor of M is given by [16]
(71)W(X,Y)Z=R(X,Y)Z-12n[S(Y,Z)X-S(X,Z)Y].
If W(X,Y)ξ=0, then (71) reduces to
(72)R(X,Y)ξ=12n[S(Y,ξ)X-S(X,ξ)Y].
Using (70) in (72), we obtain
(73)R(X,Y)ξ=(a-1)2[η(Y)X-η(X)Y].
The Weyl projective curvature tensor of M- with respect to Da-homothetic deformation is given by
(74)W-(X,Y)Z=R-(X,Y)Z-12n[S-(Y,Z)X-S-(X,Z)Y].
Now using (14) and (69) in (74), we get
(75)W-(X,Y)ξ-=R(X,Y)ξ--(a-1)2a[η(Y)X-η(X)Y].
From (73), the (75) reduces to
(76)W-(X,Y)ξ-=0.
Thus, we can state the following.Theorem 8.
LetM- be obtained from a K-contact manifold M by Da-homothetic deformation. If the Ricci tensor of M- vanishes, then it is ξ-Weyl projectively flat.



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