3_298_nagaraja.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 1, 2010, 16-25
ISSN 1307-5543 – www.ejpam.com

On N(k)-Mixed Quasi Einstein Manifolds

H G Nagaraja

Department of Mathematics, Central College, Bangalore University, Bangalore-560 001, Karnataka,

India.

Abstract. In this paper N(k)-Mixed Quasi Einstein Manifolds( N(k) − (MQE)n )are introduced and
the existence of these manifolds is proved. We give hyper surfaces of Euclidean spaces as examples of
N(k)− (MQE)n and semi symmetric, ricci symmetric and ricci recurrent N(k)− (MQE)n manifolds are
studied.

2000 Mathematics Subject Classifications: 53C25

Key Words and Phrases: N(k)-mixed quasi Einstein, mixed quasi constant curvature, ricci recurrent,
semi symmetric, ricci symmetric

1. Introduction

M.C.Chaki and R.K.Maity [1] introduced the concept quasi Einstein manifolds. A non-flat
Riemannian manifold(M n, g)(n > 2) is said to be a quasi Einstein manifold if its ricci tensor
S of type (0,2) is not identically zero and satisfies the condition

S(X , Y ) = ag(X , Y ) + bA(X )A(Y ),

where a and b are smooth functions of which b 6= 0 and A is a non zero 1-form such that
g(X , U) = A(X ), for all vector fields X and U is a unit vector field. U.C.De and Gopal Chan-
dra Ghosh [4, 5] generalized the quasi Einstein manifolds. A non-flat Riemannian manifold
(M n, g)(n > 2) is said to be a generalized quasi Einstein manifold if its ricci tensor S of type
(0,2) is not identically zero and satisfies the condition

S(X , Y ) = ag(X , Y ) + bA(X )A(Y ) + cB(X )B(Y ),

where a, b and c are certain smooth functions, A and B are non zero 1-forms, and U and
V are unit vector fields corresponding to 1-forms A and B respectively such that g(X , U) =

A(X ), g(X , V ) = B(X ) and g(U , V ) = 0. The vector fields U and V are called generators of

Email address: hgnraj�yahoo.
om (H. Nagaraja)

http://www.ejpam.com 16 c© 2009 EJPAM All rights reserved.



H. Nagaraja / Eur. J. Pure Appl. Math, 3 (2010), 16-25 17

the quasi Einstein manifold. The k-nullity distribution N(k) [8]of a Riemannian manifold M

is defined by

N(k) : p→ Np(k) = {Z ∈ TpM\R(X , Y )Z = k(g(Y, Z)X − g(X , Z)Y )}

for all X , Y ∈ T M and k is a smooth function.
M.M.Tripathy and Jeong - Jik Kim [6] introduced the notion of N(k)-quasi Einstein manifold
which is defined as follows: If the generator U belongs to the k-nullity distribution N(k), then
a quasi Einstein manifold (M n, g) is called N(k)-quasi Einstein manifold. Motivated by the
above definitions we give the following definition.

Definition 1. Let (M n, g) be a non flat Riemannian manifold. If the ricci tensor S of (M n, g) is

non zero and satisfies

S(X , Y ) = ag(X , Y ) + bA(X )B(Y ) + cB(X )A(Y ), (1)

where a, b and c are smooth functions and A and B are non zero 1-forms such that g(X , U) =

A(X ) and g(X , V ) = B(X ) for all vector fields X , and U and V being the orthogonal unit vector

fields called generators of the manifold belong to N(k), then we say that (M n, g) is a N(k)-mixed

quasi Einstein manifold and is denoted by N(k)− (MQE)n.

In this paper we introduce another notion of a manifold of mixed quasi constant curvature
similar to manifold of quasi constant curvature defined in [4]. A Riemannian manifold (M n, g)

is called a manifold of mixed quasi constant curvature if it is conformally flat and the curvature
tensor ′R of type (0,4) satisfies the condition

′R(X , Y, Z ,W ) = p[g(Y, Z)g(X ,W )− g(X , Z)g(Y,W )]

+ q
�

g(X ,W )A(Y )B(Z)− g(X , Z)A(Y )B(W ) + g(X ,W )A(Z)B(Y )

− g(X , Z)A(W )B(Y )
�

+ s[g(Y, Z)A(W )B(X )− g(Y,W )A(Z)B(X )

+ g(Y, Z)A(X )B(W )− g(Y,W )A(X )B(Z)]

(2)

Let {ei} be an orthonormal basis of the tangent space at each point of the manifold.
Taking X =W = ei and summing over i, 1≤ i ≤ n in (2), we obtain

S(Y, Z) = (n− 1)pg(Y, Z)+ (n− 1)q [A(Y )B(Z) +A(Z)B(Y )]

+s
�

2g(Y, Z)− A(Z)B(Y )− A(Y )B(Z)
�

which implies
S(Y, Z) = ag(Y, Z) + bA(Y )B(Z) + cA(Z)B(Y ) (3)

where b = c = (n− 1)q− s, a = (n− 1)p+ 2s. i.e. the space (M n, g)is mixed quasi Einstein.
Thus we have

Theorem 1. A manifold of mixed quasi constant curvature is a mixed quasi Einstein manifold.



H. Nagaraja / Eur. J. Pure Appl. Math, 3 (2010), 16-25 18

Conversely suppose (M n, g) is conformally flat mixed quasi Einstein manifold. Then

R(X , Y )Z =
1

n− 2
{g(Y, Z)QX − g(X , Z)QY + S(Y, Z)X − S(X , Z)Y }

−
r

(n− 1)(n− 2)
{g(Y, Z)X − g(X , Z)Y }.

(4)

Here Q is Ricci operator defined by S(X , Y ) = g(QX , Y ).
From the above equation, we get

′R(X , Y, Z ,W ) =g(R(X , Y )Z ,W )

=
1

n− 2

�

g(Y, Z)S(X ,W )− g(X , Z)S(Y,W )

+ S(Y, Z)g(X ,W )− S(X , Z)g(Y,W )
	

−
r

(n− 1)(n− 2)

�

g(Y, Z)g(X ,W )− g(X , Z)g(Y,W )
	

(5)

Taking X = Y = ei and taking summation over i, 1 ≤ i ≤ n in (1), we obtain r = na.
Substituting this in (5) and using (1), we get

R(X , Y, Z ,W ) = p[g(Y, Z)g(X ,W )− g(X , Z)g(Y,W )]

+ q[g(X ,W )A(Y )B(Z)− g(X , Z)A(Y )B(W ) + g(X ,W )A(Z)B(Y )

− g(X , Z)A(W )B(Y )] + s[g(Y, Z)A(W)B(X )− g(Y,W )A(Z)B(X )

+ g(Y, Z)A(X )B(W )− g(Y,W )A(X )B(Z)]

where p =
a

n− 1
, q =

b

n− 2
, s =

c

n− 2
.

i.e. (M n, g) is a manifold of mixed quasi constant curvature.

2. Existence Theorem of a N(k)-mixed Quasi Einstein Manifolds

Theorem 2. If in a conformally flat Riemannian manifold (M n, g), the ricci tensor S satisfies

the relation

S(X , Z)g(Y,W )− S(Y, Z)g(X ,W ) = β(g(Y, Z)S(X ,W )− g(X , Z)S(Y,W ) (6)

where β is a non zero scalar, then (M n, g) is a N(k)-mixed quasi Einstein manifold.

Proof. Let U be a vector field defined by g(X , U) = A(X ), ∀X ∈ T M .
Taking X =W = U in (6), we obtain

S(Y, Z) = ag(Y, Z) + bA(Y )B(Z) + cA(Z)B(Y ) (7)

where a =
−αβ

u
,α = S(U , U),u = g(U , U), b = 1

u
, c =

β

u
,

and S(U , Z) = S(Z , U) = g(QZ , U) = A(QZ) = B(Z). Therefore (M n, g) is mixed quasi Ein-
stein.



H. Nagaraja / Eur. J. Pure Appl. Math, 3 (2010), 16-25 19

If (M n, g) is conformally flat, then taking Z = U in (4), we obtain

R(X , Y )U =
1

n− 2
{A(Y )QX − A(X )QY + S(Y, U)X − S(X , U)Y } (8)

−
r

(n− 1)(n− 2)
{A(Y )X − A(X )Y } (9)

Taking β = 1 in (6), we get

S(X , Z)g(Y,W )− S(Y, Z)g(X ,W )− g(Y, Z)S(X ,W ) + g(X , Z)S(Y,W ) = 0

Taking Z = U in the above equation, we obtain

S(X , U)g(Y,W )− S(Y, U)g(X ,W )−A(Y )S(X ,W ) + A(X )S(Y,W) = 0,

which can be rewritten as g(S(X , U)Y − S(Y, U)X − A(Y )QX +A(X )QY,W ) = 0, ∀W.
Therefore we have S(X , U)Y − S(Y, U)X − A(Y )QX + A(X )QY = 0.
Substituting this in (8), we get R(X , Y )U = k(A(Y )X − A(X )Y ), where k = −r

(n−1)(n−2)
.

Therefore we have U ∈ Np(k), where k = −r

(n−1)(n−2)
.

Suppose V is a unit vector field orthogonal to U . Then, we have V ∈ Np(k).
Hence (M n, g) is a N(k)-mixed quasi Einstein manifold.

As it is well known that a 3-dimensional Riemannian manifold is conformally flat.
Thus we have

Corollary 1. A 3- dimensional manifold is N
�

−r

(n−1)(n−2)

�

-mixed quasi Einstein manifold pro-

vided (6) holds.

3. Example of a N(k)− (MQE)n manifold

Let (M n, g̃) be a hypersurface of the Euclidean space En+1. Let A be a (1,1) tensor corre-
sponding to the normal valued second fundamental tensor H.

g̃(Aξ(X ), Y ) = g(H(X , Y ),ξ) (10)

where ξ is a unit normal vector field and X and Y are tangent vector fields.
Further

Hξ(X , Y ) = g̃(Aξ(X ), Y ) (11)

The hypersurface (M n, g̃) is quasi umbilical if

Hξ(X , Y ) = α g̃(X , Y ) + βC(X )D(Y ) (12)

In view of (10), we have

H(X , Y ) = αg(X , Y )ξ+ βC(X )D(Y )ξ. (13)



H. Nagaraja / Eur. J. Pure Appl. Math, 3 (2010), 16-25 20

The Gauss equation of M n in En+1 can be written as

g̃(R̃(X , Y )Z ,W ) = g̃(H(X ,W ), H(Y, Z))− g̃(H(W, Y ), H(Z , X )) (14)

From (12) and (14), we have

′R̃(X , Y, Z ,W ) = α2 g(X ,W )g(Y, Z) +αβ g(X ,W )C(Y )D(Z)

+αβ g(Y, Z)C(X )D(W) +β2C(X )C(Y )D(W )D(Z)

−α2 g(W, Y )g(Z , X )−αβ g(W, Y )C(Z)D(X )

−αβ g(Z , X )C(W )D(Y )− β2C(W )D(Y )C(Z)D(X )

Contracting the above equation with X =W = ei and taking summation over i, 1 ≤ i ≤ n,
we obtain

S̃(Y, Z) = ag(Y, Z)+ bC(Y )D(Z)+ cC(Z)D(Y )

where a = (n− 1)α2, b = (n− 1)αβ + β2, c = −β(2α+ β).
Hence (M n, g̃) is a mixed quasi Einstein manifold.
Suppose U and V are unit orthogonal vectorfields corresponding to the 1-forms C and D

respectively. Then putting Z = U in (13), we get

H(X , U) = αC(X )ξ. (15)

Putting Z = U in (14) and using (15), we get

R̃(X , Y )U = k (C(Y )X − C(X )Y )

where k = α2. Similarly we can show that

R̃(X , Y )V = k (D(Y )X − D(X )Y )

where k = α2. Thus we have

Theorem 3. A quasi umbilical hypersurface of a Euclidean space En+1is a N(k)-mixed quasi

Einstein manifold.

4. Ricci Curvature, Eigen Vectors and Associated Scalars of a N(k)− (MQE)n

From (1) we have S(U , U) = a = S(V, V ), b = S(U , V ) = S(V, U) = c , since g(U , V ) = 0.
Therefore only one of b or c is sufficient to define a mixed quasi Einstein space. A mixed quasi
Einstein space may be defined as a Riemannian manifold in which ricci tensor S satisfies

S(X , Y ) = ag(X , Y ) + b(A(X )B(Y ) + B(X )A(Y )),

It is well known that for a unit vector field X , S(X , X ) is the ricci curvature in the direction of
X . Now if X is a unit vector field in the section spanned by U and V , then we have

1= g(X , X ) = g(αU + βV,αU + βV ) = α2 + β2,



H. Nagaraja / Eur. J. Pure Appl. Math, 3 (2010), 16-25 21

since g(U , V ) = 0 and g(U , U) = g(V, V ) = 1. Now

S(X , X ) = S(αU + βV,αU + βV )

= a+ 2bA(X )B(X ).

Thus we can state that

Theorem 4. In a N(k)− (MQE)n manifold, the ricci curvature in the direction of both U and V

is ‘a’ and the ricci curvature in all other directions of the section of U and V is a+ 2bA(X )B(X ).

Let (M n, g) be a N(k)− (MQE)n manifold.
Then S(U , U) = S(V, V ) = a from which we get g(QU , U) = g(QV, V ) = a.
Since U , V ∈ Np(k), we have,

g(R(X , Y )U ,W ) = k
�

A(Y )g(X ,W )− A(X )g(Y,W)
	

.

Putting X =W = ei and taking summation over i, 1≤ i ≤ n , we obtain

S(Y, U) = (n− 1)kA(X ) (16)

Similarly we can get
S(Y, V ) = (n− 1)kB(X ) (17)

From (1), we have
S(X , U) = aA(X )+ bB(X ) (18)

S(X , V ) = bA(X )+ aB(X ) (19)

Substracting (17) from (16) and (19) from (18), and comparing the resulting equations, we
obtain

k =
a− b

n− 1
.

Therefore
S(X , U) = (a− b)g(X , U)

and
S(X , V ) = (a− b)g(X , V ).

Therefore U and V are eigen vectors corresponding to the eigen value (a− b).

5. Semi Symmetric and Ricci Symmetric N(k)− (MQE)n Manifolds

A Riemannian manifold (M n, g) is semi symmetric if R(X , Y ).R = 0,∀X , Y ∈ T M Since U

and V are in Np(k) , we have

R(X , Y )U = k (A(Y )X − A(X )Y ) (20)

R(X , Y )V = k (B(Y )X − B(X )Y ) (21)



H. Nagaraja / Eur. J. Pure Appl. Math, 3 (2010), 16-25 22

The equation (20) is equivalent to

R(U , Y )Z = k
�

g(Y, Z)U − A(Z)Y
�

(22)

R(X , U)Z = k
�

A(Z)X − g(X , Z)U
�

(23)

The equation (21) is equivalent to

R(V, Y )Z = k
�

g(Y, Z)V − B(Z)Y
�

(24)

R(X , V )Z = k
�

B(Z)X − g(X , Z)V
�

(25)

If (M n, g) is semi symmetric then we have

R(X , Y )R(Z ,W )T − R(R(X , Y )Z ,W )T −R(Z ,R(X , Y )W )T − R(Z ,W )R(X , Y )T = 0 (26)

Putting X = U and T = V in (26), then using (21) and (22), we get

k2 �2A(Z)B(Y )W + A(W)B(Z)Y − 2B(Z)g(Y,W )U
	

= 0 (27)

From (27),we have
If k 6= 0, then 2A(Z)B(Y )W + A(W)B(Z)Y = 2B(Z)g(Y,W )U ,∀Y, Z ,W ∈ T M holds.
Putting Z = V in the above equation, we get

g(Y,W )U = A(W )Y

Taking covariant derivative on both sides of the above equation with respect to Z , we obtain

g(X , Y )∇Z U = (ZA(Y )X − A(Y ))∇Z X ,∀X , Y

Putting Y = V , we get B(X )∇Z U = 0.
Since B(X ) 6= 0, we obtain ∇Z U = 0.
i.e. U is a parallel vector field.
Similarly by taking X = V and T = U in (26),
we obtain ∇Z V = 0.
i.e. V is a parallel vector field.

Conversely suppose that U and V are parallel vector fields. Then ∇Z U = 0 and ∇Z V = 0,
which then imply that
R(X , Y )U = 0 and R(X , Y )V = 0.
Substituting this in (26) with X = U , we obtain R(U , X ).R= 0.
Similarly we get R(V, X ).R= 0.
Thus we can state that

Theorem 5. A N(k)− (MQE)n manifold with k 6= 0 satisfies R(U , X ).R = 0 (orR(V, X ).R = 0)
if and only if U (orV ) is a parallel vector field.



H. Nagaraja / Eur. J. Pure Appl. Math, 3 (2010), 16-25 23

Let (M n, g) be a N(k)− (MQE)n ricci semi symmetric manifold. Then we have

S(R(X , Y )Z ,W ) + S(Z ,R(X , Y )W = 0 (28)

Putting X = V in (28) we obtain

k
�

g(Y, Z)S(V,W )− B(Z)S(Y,W ) + g(Y,W )S(Z , V )− B(W )S(Z , Y )
	

= 0

Putting W = V in the above equation, we get

k
�

S(Z , Y )− ag(Y, Z)+ bA(Y )B(Z)− bA(Z)B(Y )
�

= 0

If k 6= 0 then we have S(Z, Y) = a g(Y, Z) - b A(Y) B(Z) + b A(Z) B(Y).
Comparing this with (1), we obtain b+ c = 0.
But we have b− c = 0, { section 4 }
Therefore b = 0 and c = 0. i.e. (M n, g) reduces to Einstein space which it is not.
Therefore we must have k = 0.
Conversely suppose k = 0. Then we obtain R(V, X )Y = 0 which implies R(V, X ).S = 0. Simi-
larly, we have, R(U , X ).S = 0. if and only if k = 0.
Thus we have,

Theorem 6. A N(k)− (MQE)n manifold satisfies R(V, X ).S = 0. and R(U , X ).S = 0 if and only

if k = 0.

6. Ricci Recurrent N(k)− (MQE)n Manifolds

Let (M n, g) be a N(k) − (MQE)n manifold. If U and V are parallel vector fields, then
∇X U = 0 and ∇X V = 0.
From which we get that R(X , Y )U = 0 and R(X , Y )U = 0. Therefore

S(X , U) = 0,S(X , V ) = 0 (29)

From (1), we have

S(X , U) = aA(X )+ bB(X )and (30)

S(X , V ) = aB(X ) + bA(X ) (31)

From (29), (30) and (31), we have a = b.
Therefore we can rewrite the equation (1) in the following form:

S(X , Y ) = a
�

g(X , Y ) + A(X )B(Y ) + B(X )A(Y )
	

.

Taking the covariant derivative of the above equation with respect to Z , we obtain

∇ZS(X , Y ) = da(Z)
�

g(X , Y ) +A(X )B(Y ) + B(X )A(Y )
	



REFERENCES 24

since ∇X U = 0 and ∇X V = 0 imply that ∇ZA(X ) = 0 and ∇Z B(X ) = 0.

Therefore (∇ZS)(X , Y ) =
da(Z)

a
S(X , Y ),

i.e. the manifold (M n, g) is ricci recurrent.
Conversely, suppose that N(k)− (MQE)n manifold is ricci recurrent. Then

�

∇X S
�

(Y, Z) = D(X )S(Y, Z), D(X ) 6= 0.

But
�

∇X S
�

(Y, Z) = X S(Y, Z)− S
�

∇X Y, Z
�

− S
�

Y,∇X Z
�

Therefore
D(X )S(Y, Z) = X S(Y, Z)− S

�

∇X Y, Z
�

− S
�

Y,∇X Z
�

Putting Y = Z = U , we obtain
X a− aD(X ) = 2a

�

g
�

∇X U , U
�

+ B(∇X U)
�

i.e. (da− aD)X = 2aB(∇X U). since g(U , U) = 1 implies g(∇X U , U) = 0
Therefore B(∇X U) = 0 if and only if

(da)(X ) = aD(X ) (32)

But B(∇X U) = 0 implies that either U is a parallel vector field or ∇X U ⊥ V.
Similarly we have, if (32) holds then either V is a parallel vector field or ∇X V ⊥ U .
Thus we can state that

Theorem 7. A N(k)(MQE)n manifold, where the generators U and V are parallel is a ricci

recurrent manifold.Conversely suppose that N(k) − (MQE)n manifold is ricci recurrent, then

either the vector field U ( or V) is parallel or ∇X U ⊥ V ( or ∇X V ⊥ U).

References

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297-306.

[2] B.Y.Chen, Geometry of Submanifolds, Marcel Dekker, Inc., New York, 1973.

[3] B.Y.Chen and K. Yano, hypersurfaces of a conformally flat space, Tensor, N.S., 26 (1972)
318-322.

[4] U.C.De and Gopal Chandra Ghosh, On quasi Einstein manifolds, Periodica Mathematica
Hungarica, 48(1-2)(2004) 223-231.

[5] U.C.De and Gopal Chandra Ghosh, On generalized quasi Einstein manifolds, Kyungpook
math. J., 44(2004) 607-615.

[6] Mukut Mani tripathi and Jeong-Sik Kim, On N(k)-quasi Einstein manifolds, Commun.
Korean Math. Soc. 22(3) (2007),411-417.



REFERENCES 25

[7] Özgür Cihan and Sular Sibel, On N(k)-quasi Einstein manifolds satisfying certain condi-

tions, Balkan J. Geom. Appl. 13(2008), no: 2, 74-79.

[8] S.Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J. 40 (1988)
441-448.