Characterization on N(k)-Mixed Quasi-Einstein Manifold.

1. Introduction

A Riemannian manifold (M, g) with dimension (n [greater than or equal to] 2) is said to be an Einstein manifold if the Ricci tensor satisfies the condition S(X, Y) = [r/n] g(X, Y), holds on M, here S and r denote the Ricci tensor and the scalar curvature of (M, g) respectively. According to  the above equation is called the Einstein metric condition. Einstein manifolds play an important role in Riemannian Geometry, as well as in general theory of relativity. The notion of quasi-Einstien manifold were defined in (,,,,,). A non-flat Riemannian manifold (M, g), (n [greater than or equal to] 2) is said to be an quasi Einstein manifold if its Ricci tensor S of type (0, 2) satisfies the condition

S(X, Y)=ag(X,Y)+bA(X)A(Y) (1.1)

where a and b are scalars of which b [not equal to] 0 and A is non-zero 1-form such that g(X, U) = A(X) for all vector field X and U is a unit vector field.

The notion of quasi Einstein manifold was introduced in a paper  by M.C.Chaki and R.K.Maity. According to them a non-flat Riemannian manifold ([M.sup.n], g),(n [greater than or equal to] 3) is defined to be a quasi Einstein manifold if its Ricci tensor S of type (0, 2) satisfies the condition

S(X, Y) = ag(X, Y) + bA(X)A(Y) (1.2)

and is not identically zero, where a, b are scalars b [not equal to] 0 and A is a non-zero 1-form such that

g(X, U) = A(X), [for all]X [member of] TM. (1.3)

U being a unit vector field.

In such a case a, b are called the associated scalars. A is called the associated 1-form and U is called the generator of the manifold. Such an n-dimensional manifold is denoted by the symbol[(QE).sub.n].

Again, U.C.De and G.C.Ghosh defined generalized quasi Einstein manifold . A non-flat Riemannian manifold is called a generalized quasi Einstein manifold (, , , , , , )if its Ricci-tensor S of type (0, 2) is non-zero and satisfies the condition

S(X, Y) = ag(X, Y) + bA(X)A(Y) + cB(X)B(Y) (1.4)

where a, b, c, are non-zero scalars and A, B are two 1-forms such that

g(X, U) = A(X) and g(X, V) = B(X) (1.5)

U, V being unit vectors which are orthogonal, i.e,

g(U, V) = 0. (1.6)

The vector fields U and V are called the generators of the manifold. This type of manifold are denoted by G[(QE).sub.n].

The k-nullity distribution  of a Riemannian manifold M is defined by

N(k) : p [right arrow] [N.sub.p](k) = {Z E [T.sub.p]M \ R(X, Y)Z = k(g(Y, Z)X - g(X, Z)Y)}. (1.7)

for all X,Y E TM and k is a smooth function. M.M.Tripathi and Jeong jik kim  introduced the notion of N(k)-quasi Einstein manifold which defined as follows: If the generator U belongs to the k-nullity distribution N(k), then a quasi Einstein manifold ([M.sup.n],g) is called N(k)-quasi Einstein manifold.

In , H.G. Nagaraja introduced the concept of N(k)-mixed quasi Einstein manifold and mixed quasi constant curvature.A non flat Riemannian manifold ([M.sup.n], g) is called a N(k)-mixed quasi Einstein manifold  if its Ricci tensor of type (0, 2) is non zero and satisfies the condition

S(X, Y) = ag(X, Y) + bA(X)B(Y) + cB(X)A(Y), (1.8)

where a, b, c, are smooth functions and A, B are non zero 1-forms such that

g(X, U) = A(X) and g(X, V) = B(X) V X, (1.9)

U, V being the orthogonal unit vector fields called generators of the manifold be-long to N(k).Such aa manifold is denoted by the symbol N(k) - [(MQE).sub.n]. Again a Riemannian manifold ([M.sup.n],g) is called of mixed quasi constant curvature  if it is conformally flat and 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)]. (1.10)

Let M be an m-dimensional, m [greater than or equal to] 3, Riemannian manifold and p E M. Denote by K([pi]) or K(U [and] V) the sectional curvature of M associated with a plane section [pi] [subset.bar] [T.sub.p]M, where {U, V} is an orthonormal basis of [pi]. For a ndimensional subspace L [subset.bar] [T.sub.p]M, 2 [less than or equal to] n [less than or equal to] m, its scalar cuvrvature [tau](L) is denoted by [mathematical expression not reproducible], where {[e.sub.1],[e.sub.2],...,[e.sub.n]} is any orthonormal basis of

L().

In  the result for odd dimensional Einstein spaces was obtained by Dumitru. Also in  Bejan generalized these results (both odd and even dimensions)to quasi Einstein manifold. Also characterization of super quasi-Einstein manifold for both of odd and even dimensions was studied in . From above studies, we have given characterization of N(k)-mixed quasi-Einstein manifold for both of odd and even dimensions. Next we obtain that a N(k)-mixed quasi-Einstein manifold is semi mixed quasi-Einstein manifold if either of generators is parallel vector field.

Geodesic mappings of Einstein spaces were studied in (,,,,,).

2. Characterization of N(k)-mixed quasi-Einstein manifold manifold

In this section we establish the characterization of odd and even dimensional N(k) - [(MQE).sub.n].

Theorem 2.1. A Riemannian manifold of dimension (2n + 1) with n [greater than or equal to] 2 is N(k)-mixed quasi-Einstein manifold if and only if the Ricci operator Q has eigen vector fields U and V such that at any point p [member of] M, there exist three real numbers [alpha] and [beta] satisfying

[tau](P) + [alpha] = [tau]([P.sup.[perpendicular to]]); U, V [member of] [T.sub.p][P.sup.[perpendicular to]], [tau](N) + [alpha] = [tau]([N.sup.[perpendicular to]]); U [member of] [T.sub.p]N, V [member of] [T.sub.p][N.sup.[perpendicular to]], [tau](R) + [beta] = [tau]([R.sup.[perpendicular to]]); U [member of] [T.sub.p]R, V [member of] [T.sub.p][R.sup.[perpendicular to]],

for any n-plane sections P, N and (n+ 1)-plane section R where [P.sup.[perpendicular to]], [N.sup.[perpendicular to]] and [R.sup.[perpendicular to]] denote the orthogonal complements of P, N and R in [T.sub.p]M respectively and

[alpha] = a/2, [beta] = - c/2,

where a, b, c are scalars.

Proof. First suppose that M is a (2n+1) dimensional N(k)-mixed quasi-Einstein manifold, so

S(X, Y) = ag(X, Y) + bA(X)B(Y) + cA(Y)B(X), (2.1)

where a,b,c are scalars such that b, c are nonzero and A, B are two nonzero 1-forms such that g(X, U) = A(X) and g(X, V) = B(X), [for all]X G X(M), U, V being unit vectors which are orthogonal, i.e., g(U,V) = 0.

Let P [subset.bar] [T.sub.p]M be an n-dimensional plane orthogonal to U, V and let {[e.sub.1], [e.sub.2],..., [e.sub.n]} be orthonormal basis of it. Since U and V are orthogonal to P, we can take orthonormal basis {[e.sub.n+1], [e.sub.n+2],..., [e.sub.2n+1]} of [P.sup.[perpendicular to]] such that [e.sub.2n] = U and [e.sub.2n+1] = V. Thus {[e.sub.1], [e.sub.2],..., [e.sub.n], [e.sub.n+1], [e.sub.n+2],..., [e.sub.2n+1]} is an orthonormal basis of [T.sub.p]M.Then we can take X = Y = ei in (2.1), we have

[mathematical expression not reproducible]

By use of (2.1) for any 1 [less than or equal to] i [less than or equal to] 2n + 1, we can write

S([e.sub.1], [e.sub.1]) = K([e.sub.1] [and] [e.sub.2])+K([e.sub.1][and][e.sub.3]) + ... +K([e.sub.1][and][e.sub.2n-1])+K([e.sub.1][and]U) + K ([e.sub.1] [and]V) = a, S([e.sub.2], [e.sub.2]) = K([e.sub.2] [and] [e.sub.1]) + K([e.sub.2] [and][e.sub.3]) + ...+K([e.sub.2][and][e.sub.2n-1])+K([e.sub.2][and]U)+K([e.sub.2][and]V) = a, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S ([e.sub.2n-1], [e.sub.2n-1]) K ([e.sub.2n-1][and][e.sub.1])+K ([e.sub.2n-1][and][e.sub.2])+K ([e.sub.2n-1][and] [e.sub.3])+...+K ([e.sub.2n-1][and] V) a, S(U, U) = K(U [and] [e.sub.1]) + K(U [and] [e.sub.2]) + ... + K(U [and] [e.sub.2n-1]) + K(U [and] V) = a, S(V, V) = K(V [and] [e.sub.1]) + K(V [and] [e.sub.2]) + ... + K(V [and] [e.sub.2n-1]) + K(V [and]U) = a.

Adding first n-equations, we get

[mathematical expression not reproducible] (2.2)

Then adding the last (n + 1)-equations, we have

[mathematical expression not reproducible] (2.3)

Then, by substracting the equation (2.2) and (2.3), we obtain

[tau]([P.sup.[perpendicular to]]) - [tau](P) = a/2.

Therefore [tau](P) + [alpha] = [tau]([P.sup.[perpendicular to]]), where,

[alpha] = a/2.

Similarly, Let N [subset.bar] [T.sub.p]M be an n-dimensional plane orthogonal to V and let {[e.sub.1], [e.sub.2], . . ., [e.sub.n]} be orthonormal basis of it. Since V is orthogonal to N, we can take an orthonormal basis {[e.sub.n+1], [e.sub.n+2], . . . ,[e.sub.2n+1]} of [N.sup.[perpendicular to]] orthogonal to U, such that [e.sub.n] = U and [e.sub.2n+1] = V, respectively. Thus, {[e.sub.1], [e.sub.2], . . ., [e.sub.n], [e.sub.n+1], [e.sub.n+2], . . ., [e.sub.2n+1]} is an orthonormal basis of [T.sub.p]M. Then we can take X = Y = ei in (2.1) to have

[mathematical expression not reproducible]

Adding first n-equations, we get

[mathematical expression not reproducible] (2.4)

and adding the last (n + 1)-equations, we have

[mathematical expression not reproducible] (2.5)

Then, by substracting the equation (2.4) and (2.5), we obtain

[tau]([N.sup.[perpendicular to]]) - [tau](N) = a/2.

Therefore [tau](N) + [alpha] = [tau]([N.sup.[perpendicular to]]), where,

[alpha] = a/2.

Analogously, Let R [subset.bar] [T.sub.p]M be an (n+1)-plane orthogonal to V and let {[e.sub.1], [e.sub.2], . . ., [e.sub.n+1]} be orthonormal basis of it. Since V is orthogonal to R, we can take an orthonormal basis {[e.sub.n+2], [e.sub.n+3],..., [e.sub.2n], [e.sub.2n+1]} of [R.sup.[perpendicular to]] orthogonal to U, such that [e.sub.n+1] = U and [e.sub.2n+1] = V. Thus, {[e.sub.1], [e.sub.2], . . ., [e.sub.n], [e.sub.n+1], [e.sub.n+2], . . ., [e.sub.2n+1]} is an orthonormal basis of [T.sub.p]M. Then we can take X = Y = [e.sub.i] in (2.1) to have

[mathematical expression not reproducible]

Adding the first (n + 1)-equations, we get

[mathematical expression not reproducible] (2.6)

and adding the last n-equations, we have

[mathematical expression not reproducible] (2.7)

Then, by substracting the equation (2.6) and (2.7), we obtain

[tau]([R.sup.[perpendicular to]]) - [tau](R) = -a/2.

Therefore [tau](R) + [beta] = [tau]([R.sup.[perpendicular to]]), where,

[beta] = -a/2.

Conversely, let [V.sub.1] be an arbitrary unit vector of [T.sub.p]M, at p [member of] M, orthogonal to U and V. We take an orthonormal basis {[e.sub.1], [e.sub.2], . . ., [e.sub.n], [e.sub.n+1], [e.sub.n+2], . . ., [e.sub.2n+1]} of [T.sub.p]M such that [V.sub.1] = [e.sub.1], [e.sub.n+1] = U and [e.sub.2n+1] = V. We consider n-plane section N and (n + 1)-plane section R in [T.sub.p]M as follows N = span {[e.sub.2], . . ., [e.sub.n], [e.sub.n+1]} and R = span {[e.sub.1], [e.sub.2], . . ., [e.sub.n], [e.sub.n+1]} respectively.Then we have [N.sup.[perpendicular to]] = span {[e.sub.1], [e.sub.n+]2, . . ., [e.sub.2]n, [e.sub.2n+1]} and R = span {[e.sub.n+2]. . . ., [e.sub.2n]} respectively. Now

[mathematical expression not reproducible]

Therefore, S([V.sub.1],[V.sub.1]) = [alpha] - [beta], for any unit vector [V.sub.1] [member of] [T.sub.p]M, orthogonal to U and V. Then we can write for any 1 [less than or equal to] i [less than or equal to] 2n + 1, S([e.sub.i],[e.sub.i]) = [alpha] - [beta], since S([V.sub.1],[V.sub.1]) = ([alpha] - [beta])g([V.sub.1], [V.sub.1]). It follows that S(X, X) = ([alpha] - [beta])g(X,X) and S(Y,Y) = ([alpha] -[beta])g(Y,Y)+ [K.sub.1]A(Y)B(Y)+[K.sub.2]B(Y)A(Y) for any X [member of] [[span{U}].sup.[perpendicular to]] and Y [member of] [[span{V}].sup.[perpendicular to]], where A, B are the dual forms of U and V with respect to g, respectively and [K.sub.1], [K.sub.2] are scalars, such that [K.sub.1] [not equal to] 0, [K.sub.2] [not equal to] 0.

Now from the above equations, we get from symmetry that S with tensors ([alpha]-[beta])g and ([alpha]-[beta])+[K.sub.1](A[cross product]B)+[K.sub.2](A[cross product]B) must coincide on the complement of U and V, respectively, that is S(X, Y) = ([alpha] - [beta])g(X, Y) + [K.sub.1]A(X)B(Y) + [K.sub.2]B(X)A(Y), for any X,Y [member of] [[span{U, V}].sup.[perpendicular to]] . Since U and V are eigenvector fields of Q, we also have S(X, U) = 0 and S(Y, V) = 0 for any X,Y [member of] [T.sub.p]M orthogonal to U and V. Thus, we can extend the above equation to

S(X, Z) = ([alpha] - [beta])g(X, Z) + [K.sub.1]A(X)B(Z) + [K.sub.2]A(Z)B(X) (2.8)

, for any X [member of] [[span{U, V}].sup.[perpendicular to]] and Z [member of] [T.sub.p]M, where [K.sub.1], [K.sub.2], are scalars and [K.sub.1] = 0, [K.sub.2] [not equal to] 0,. Now, let us consider the n-plane section P and (n + 1)-plane section R in [T.sub.p]M as follows P = span {[e.sub.1], [e.sub.2], . . ., [e.sub.n]} and R = span {[e.sub.1], [e.sub.2], . . ., [e.sub.n], U}. Then we have [P.sup.[perpendicular to]] = span {U, [e.sub.n+2], . . ., [e.sub.2n+1]} and [R.sup.[perpendicular to]] = span {[e.sub.n+2], . . ., [e.sub.2n], [e.sub.2n+1]} respectively. Now

[mathematical expression not reproducible].

Therefore we can write

S(U, U) = ([alpha] - [beta])g(U, U). (2.9)

Analogously, let us consider the n-plane section P and N [member of] [T.sub.p]M as follows P = span {[e.sub.1], [e.sub.2], . . ., [e.sub.n]} and N = span {[e.sub.n+1], [e.sub.n+2], . . ., [e.sub.2n]} respectively. Then we have [P.sup.[perpendicular to]] = span {[e.sub.n+1], [e.sub.n+2], . . . , [e.sub.2n], V} and [N.sup.[perpendicular to]] = span {[e.sub.1],..., [e.sub.n], V} re-spectively. Now, we have

[mathematical expression not reproducible]

Then, we get

S(V, V) = 2[alpha]g(V, V) + [K.sub.1]A(V)B(V) + [K.sub.2]A(V)B(V). (2.10)

Now from (2.8), (2.9) and (2.10) we can write the Ricci tensor by

S(X, Y) = [[lambda].sub.1]g(X, Y)+[K.sub.1]A(X)B(Y)+[K.sub.2]B(X)A(Y), (2.11)

for any X, Y [member of] [T.sub.p]M. From (2.11) it follows that M is a N(k)-mixed quasi-Einstein manifold, where [lambda]1,[K.sub.1],[K.sub.2], are scalars and [K.sub.1] = 0,[K.sub.2] = 0,. Hence the theorem is proved.

Theorem 2.2. A Riemannian manifold of dimension 2n with n [greater than or equal to] 2 is N(k)-mixed quasi-Einstein manifold if and only if the Ricci operator Q has eigen vector fields U and V such that at any point p G M, satisfying

[tau](P) = [tau]([P.sup.[perpendicular to]]); U, V [member of] [T.sub.p][P.sup.[perpendicular to]], [tau](N) = [tau]([N.sup.[perpendicular to]]); U [member of] [T.sub.p]N, V [member of] [T.sub.p][N.sup.[perpendicular to]], [tau](R) = [tau]([R.sup.[perpendicular to]]); U [member of] [T.sub.p]R, V [member of] [T.sub.p][R.sup.[perpendicular to]],

for any n-plane section P, N and (n + 1)-plane section R where [P.sup.[perpendicular to]], [N.sup.[perpendicular to]] and [R.sup.[perpendicular to]] denote the orthogonal complements of P, N and R in [T.sub.p]M respectively.

Proof. Let P and R be n-plane sections and N be an (n - 1)-plane section such that, P = span{[e.sub.1], [e.sub.2], . . ., [e.sub.n]}, R = span{[e.sub.n+1], [e.sub.n+2], . . ., [e.sub.2n]} and N = span{[e.sub.2], [e.sub.3], . . ., [e.sub.n]} respectively. Therefore the orthogonal complements of these sections can be written as [P.sup.[perpendicular to]] = span{[e.sub.n+1], [e.sub.n+2], . . ., [e.sub.2n]}, [R.sup.[perpendicular to]] = span{[e.sub.1], [e.sub.2], . . ., [e.sub.n]} and [N.sup.[perpendicular to]] = span{[e.sub.1], [e.sub.n+1],..., [e.sub.2n]}.

Then rest of the proof is similar to the proof of Theorem 2.1.

3. N(k) - [MQE.sub.n] with the parallel vector field generators

Theorem 3.1. A N(k)- mixed quasi-Einstein manifold is semi-mixed quasi-Einstein manifold if either of generators is parallel vector field.

Proof. By the definition of the Riemannian curvature tensor, if U is parallel vector field, then we find that

R(X, Y)U = [[nabla].sub.X][[nabla].sub.Y] U - [[nabla].sub.Y] [[nabla].sub.X]U - [[nabla].sub.[X,Y]]U = 0,

and consequently we get

S(X, U) = 0. (3.1)

Again, putting Y = U in the equation (1.8) and applying (1.9) , we have

S(X, U) = ag(X, U) + cg(X, V).

So, if U is a parallel vector field, by (3.1), we get

ag(X, U) + cg(X, V) = 0. (3.2)

Now, putting X = V in the equation (3.2) and using (1.9) we get c = 0. So, if U is parallel vector field in a mixed-quasi-Einstein manifold, then the manifold is semi-mixed quasi Einstein manifold.

Again, if V is parallel vector field, then R(X, Y)V = 0. Contracting, we get

S(Y, V) = 0. (3.3)

Putting X = V in the equation (1.8) and applying (1.9), we obtain

S(Y, V) = ag(Y, V) + bg(Y, U).

If, V is a parallel vector field, by (3.3), we get

ag(Y,V)+bg(Y,U)=0. (3.4)

Putting Y = U and using (3.4), (1.9), we obtain b = 0, i.e., the manifold is semi mixed quasi-Einstein manifold.

References

 Blair, D. E: Contact manifolds in Riemannian geometry, Lecture notes on Mathematics 509,(1976) springer verlag.

 Bhattacharyya, A and De, T: On mixed generalized quasi-Einstein manifold, Diff.geometry and Dynamical system ,Vol. 2007, pp. 40-46.

 Bhattacharyya, A and Debnath, D: On some types of quasi Einstein manifolds and generalized quasi Einstein manifolds, Ganita, Vol. 57, no. 2, 2006, 185-191.

 Bhattacharyya, A; De, T and Debnath, D: On mixed generalized quasi-Einstein manifold and some properties,An. St. Univ. "Al.I.Cuza"Iasi S.I.a Mathematica (N.S)53(2007),No.1, 137-148.

 Bhattacharyya, A; Tarafdar, M and Debnath, D: On mixed super quasi-Einstein manifold, Diff.geometry and Dynamical system ,Vol.10, 2008, pp. 44-57.

 Bhattacharyya, A and Debnath, D: Some types of generalized quasi Einstein, pseudo Ricci- symmetric and weakly symmetric manifold, An. St. niv. "Al.I.Cuza" Din Iasi(S.N)Mathematica, Tomul LV, 2009, f.1145-151.

 Bejan, C.L: Characterization of quasi Einstein manifolds, An. Stiint. Univ."Al.I.Cuza" Iasi Mat. (N.S.), 53(2007), suppl.1, 67-72.

 Besse, A. L.: Einstein manifolds., New York, Springer-Verlag. 1987.

 Chaki, M.C and Ghosh, M.L: On quasi conformally flat and quasiconformally conservative Riemannian manifolds,An. St. Univ. "AL.I.CUZA"IASI Tomul XXXVIII S.I.a Mathematica f2 (1997),375-381.

 Chaki, M.C and Maity, R.K: On quasi Einstein manifold, publ. Math.Debrecen 57(2000), 297-306.

 Debnath, D: On N(k) Mixed Quasi Einstein Manifolds and Some global properties, Acta Mathematica Academiae Paedagogicae Nyregyhziensis, accepted in vol. 33(2), 2017.

 De,U. C.and Ghosh,G. C: On generalized quasi-Einstein manifolds, Kyungpook Math. J., 44(2004), 607-615.

 Dumitru, D: On Einstein spaces of odd dimension, Bul. Transilv. Univ. Brasov Ser. B (N.S.), 14, No.49, 2007 suppl., 95-97.

 Deszcz, R; Glogowska,M; Holtos, M and Senturk, Z: On certain quasi-Einstein semisymmetric hypersurfaces., Ann. Univ. Sci. Budapest. Eotovos Sect. Math, 41 (1998), 151-164 (1999).

 Debnath, D and Bhattacharyya, A: Some global properties of mixed super quasi-Einstein manifold, Diff.geometry and Dynamical system ,Vol.11, 2009, pp. 105-111.

 De, U.C and Ghosh, G.C: On quasi Einstein manifolds, Periodica Mathematica Hungarica Vol. 48(1-2). 2004, pp. 223-231.

 Debnath, D and Bhattacharyya, A: On some types of quasi Eiestein, generalized quasi Einstein and super quasi Einstein manifolds, J.Rajasthan Acad. Phy. Sci, Vol. 10, No.1 March 2011, PP-33-40.

 Debnath, D: Some properties of mixed generalized and mixed super quasi Einstein manifolds, Journal of Mathematics, Vol.II, No.2(2009),pp-147-158.

 Formella,S and Mikes, J: Geodesic mappings of Einstein spaces, Szczecinske rocz. naukove, Ann. Sci. Stetinenses, 9(1994),31-40.

 Halder, K; Pal,B; Bhattacharya,A; De,T: Characterization of super quasi Einstein manifolds, An.Stiint. Univ."Al.I.Cuza" Iasi Mat.(N.S.),60, No. 1, 2014, 99-108.

 Hinterleiner, I and Mikes, J: Geodesic mappings and Einstein spaces, In: Geometric methods in physics, Trends in Mathematics, Birkhauser, Basel 2013, 331-335.

 Mikes, J: Geodesic mappings of affine-connected and Riemannian spaces, J. Math. Sci. (New York) 78, No.3, 1996, 311-333.

 Mikes, J: Geodesic mappings of special Riemannian spaces, Coll. Math. Soc. J. Bolyai. 46. Topics in Diff. Geom. Debrecen (Hungary), 1984, Amsterdam (1988), 793-813.

 Mikes, J: Geodesic mappings of Einstein spaces, Math. Notes, 28(1981), 922-923.

 Mikes, J at al.: Differential geometry of special mappings, Palacky Univ. Press, Olomouc. 2015.

 Nagaraja, H.G: On N(k)-mixed quasi Einstein manifolds, European Journal of Pure and Applied Mathematics. Vol. 3, No. 1, 2010, 16-25

 O'Neill,B: Semi-Riemannian Geometry wih Applications to Relativity, Pure and Appl. Math., Academic Press Inc., New York, 1983.

 Tripathi, M.M and Kim,Jeong Sik: On N(k)- quasi Einstein manifolds, Commun. Korean Math. Soc. 22(3) (2007), 411-417.

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

Dipankar Debnath

Department of Mathematics, Bamanpukur High School, Bamanpukur,PO-Sree Mayapur, West Bengal,India,PIN-741313 Email:dipankardebnat[h.sub.1]23@gmail.com

Arindam Bhattacharyya

Department of Mathematics Jadavpur University, Kolkata-700032, India. E-mail: bhattachar1968@yahoo.co.in

Received May 26, 2017 Accepted December 14, 2017
COPYRIGHT 2017 Aletheia University
No portion of this article can be reproduced without the express written permission from the copyright holder.
Author: Printer friendly Cite/link Email Feedback Debnath, Dipankar; Bhattacharyya, Arindam Tamsui Oxford Journal of Information and Mathematical Sciences Report 1USA May 1, 2017 4119 Entropy based Multi-objective Matrix Game Model with Fuzzy Goals. AN EXTREME POINT THEOREM ON HYPERSPACE. Manifolds (Mathematics)