A CHARACTERIZATION OF GENERALIZED QUASI-EINSTEIN MANIFOLDS

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1 Novi Sad J. Math. Vol. 4, No. 1, 01, A CHARACTERIZATION OF GENERALIZED QUASI-EINSTEIN MANIFOLDS Dan Dumitru 1 Abstract. The aim of this paper is to give a characterisation of generalized quasi-einstein manifolds in terms of scalar curvatures of subspaces of the tangent space. AMS Mathematics Subject Classification (010): 53C15, 53B0, 53C4 Key words and phrases: quasi-einstein manifold, generalized quasi-einstein 1. Introduction According to ([]) we have the following definition. Definition 1.1. A non-flat Riemannian manifold (M, g), n >, is said to be a quasi-einstein manifold if its Ricci tensor Ric of type (0, ) is not identically zero and satisfies the condition Ric(X, Y ) = ag(x, Y ) + ba(x)a(y ) for every X, Y Γ(T M), where a, b are real scalars, b 0 and A is a non-zero 1-form on M, such that A(X) = g(x, U) for all vector field X Γ(T M), U being a unit vector field which is called the generator of the manifold. According to ([4]) we have the following definition. Definition 1.. A non-flat Riemannian manifold (M, g), n >, is said to be a generalized quasi-einstein manifold if its Ricci tensor Ric of type (0, ) is not identically zero and satisfies the condition Ric M (X, Y ) = ag(x, Y ) + ba(x)a(y )+cb(x)b(y ) for every X, Y Γ(T M), where a, b, c are real scalars, b 0, c 0 and A, B are non-zero 1-form on M such that A(X) = g(x, U), B(X) = g(x, V ), g(u, V ) = 0 for all vector field X Γ(T M), U, V being unit vector fields which are called the generators of the manifold. Let M be a Riemannian n-manifold. Let p M and L T p M a subspace of dimension r n. Let {e 1,..., e r } be a basis for L. We will denote by τ(l) = K(e i e j ), where K(e i e j ) is the sectional curvature of 1 i<j r the plane determinated by {e i, e j }. τ(l) is called the scalar curvature of L. In these conditions, the orthoganal complement of L is the plane spanned by {e r+1,..., e n } and is denoted by L. We give now some characterisations of Einstein-type manifolds. 1 Spiru Haret University of Bucharest, Department of Mathematics and Computer Science, 13 Ghica str., Bucharest, Romania, dandumitru1984@yahoo.com

2 90 Dan Dumitru Theorem 1.1. ([6]) Let M be a Riemannian 4-manifold. Then M is an Einstein space if and only if K(π) = K(π ) for any plane section π T p M, where π denotes the orthogonal complement of π in T p M for every p M. Theorem 1.. ([3]) Let M be a Riemannian (n)-manifold. Then M is an Einstein space if and only if τ(l) = τ(l ) for any n-plane section L T p M, where L denotes the orthogonal complement of L in T p M for every p M. Theorem 1.3. ([5]) Let M be a Riemannian (n + 1)-manifold. Then M is an Einstein space of constant λ if and only if τ(l) + λ = τ(l ) for any n-plane section L T p M, where L denotes the orthogonal complement of L in T p M for every p M. Theorem 1.4. ([1]) Let (M, g) be a Riemannian (n + 1)-manifold with n. Then M is quasi-einstein if and only if the Ricci operator Ric has an eigenvector ξ such that at any p M, there exist two real numbers a, b satisfying τ(p )+a = τ(p ) and τ(n) + b = τ(n ) for any n-plane section P and (n + 1)-plane section N, both orthogonal to ξ in T p M, where P and N denote respectively the orthogonal complement of P and N in T p M. Theorem 1.5. ([1]) Let (M, g) be a Riemannian (n)-manifold with n. Then M is quasi-einstein if and only if the Ricci operator Ric has an eigenvector ξ such that at any p M, there exists the real number c satisfying τ(p ) + c = τ(p ) for any n-plane section P orthogonal to ξ in T p M, where P denotes the orthogonal complement of P in T p M.. Main results The aim of this paper is to extend in some sense the results from ([1, 3, 5, 6]) to the case when the ambient space is a generalized quasi-einstein space. Thus, we will give a characterization of generalized quasi-einstein manifolds in terms of scalar curvatures of n-planes included in the tangent space. Theorem.1. Let M be a Riemannian (n + 1)-manifold, n. Then the following conditions are equivalent: 1) M is a generalized quasi-einstein manifold with Ric(X, Y ) = ag(x, Y ) + ba(x)a(y )+ cb(x)b(y ) for every X, Y Γ(T M), where a, b, c are real scalars and A, B are non-zero 1-forms on M such that A(X) = g(x, U), B(X) = g(x, V ), g(u, V ) = 0 for all vector field X Γ(T M), U, V being unit vector fields. ) a) τ(l 1 ) = τ(l 1 ) + 1 (a + b + c) for any n-plane section L 1 T p M such that U, V / L 1, b) τ(l ) = τ(l ) + 1 (a b c) for any n-plane section L T p M such that U, V L, c) τ(l 3 ) = τ(l 3 ) + 1 (a + b c) for any n-plane section L 3 T p M such that U / L 1, V L 3, d) τ(l 4 ) = τ(l 4 ) + 1 (a b + c) for any n-plane section L 4 T p M such that U L 4, V / L 4, where L denotes the orthogonal complement of L in T p M for every p M.

3 A characterization of generalized quasi-einstein manifolds 91 Proof. 1) ). Let p M and {e 1,.., e n,..., e n+1 } an orthonormal frame of T p M such that U = e 1 and V = e. We know that Ric(X, Y ) = R(X, e i, Y, e i ) = ag(x, Y ) + ba(x)a(y ) + cb(x)b(y ) n+1 i=1 for every X, Y Γ(T M). Let X = Y = e i. This implies that Ric(e i ) = Ric(e i, e i ) = n+1 R(e i, e j, e i, e j ) = a for every i {3,..., n + 1}. In the same j=1 way we obtain that Ric(U) = Ric(e 1 ) = a + b and Ric(V ) = Ric(e ) = a + c. We will write now all the equations and by the formula of Ricci cuvature, we will have the following system of n + 1 equations: n+1) 1) Ric(e 1 ) = K(e 1 e ) + K(e 1 e 3 ) K(e 1 e n+1 ) = a + b ) Ric(e ) = K(e e 1 ) + K(e e 3 ) K(e e n+1 ) = a + c 3) Ric(e 3 ) = K(e 3 e 1 ) + K(e 3 e ) K(e 3 e n+1 ) = a. Ric(e n+1 ) = K(e n+1 e 1 ) + K(e n+1 e )+...+K(e n+1 e n ) = a Without any loss of generality we will consider the following n-planes from T p M : L 1 = Sp({e 3, e 4,..., e n+ }), L = Sp({e 1, e,..., e n }), L 3 = Sp({e, e 3,..., e n+1 }), L 4 = Sp({e 1, e 3, e 4,..., e n+1 }). Now, by summing the first n equations we have the following relation: (i) τ(l ) + 1 i n<j n+1 K(e i e j ) = na + b + c By summing the last n + 1 equations we have another relation: (ii) τ(l ) + Then (ii) (i) implies: 1 j n<i n+1 a b c =τ(l ) τ(l ) + K(e i e j ) = (n + 1)a 1 j n<i n+1 τ(l ) = τ(l ) + 1 (a b c). K(e i e j ) 1 i n<j n+1 K(e i e j ) In a similar way, by summing the equations from 3) to n+) we have:

4 9 Dan Dumitru (iii) τ(l 1 ) + K(e i e j ) + K(e i e j ) = na 3 i n+<j n+1 3 i n+, j {1,} Also, by summing the remaining equations we have: (iv) τ(l 1 ) + K(e i e j ) + K(e i e j ) = (n + 1)a + b + c 3 i n+<j n+1 3 i n+, j {1,} Then (iv) (iii) implies: a + b + c = τ(l 1 ) τ(l 1 ) τ(l 1 ) = τ(l 1 ) + 1 (a + b + c). In a similar way, by summing the equations from ) to n+1) we have: (v) τ(l 3 ) + i n+1<j n+1 K(e i e j ) + K(e i e 1 ) = na + c i n+1 Also, by summing the remaining equations we have: (vi) τ(l 3 ) + i n+1<j n+1 Then (vi) (v) implies: K(e i e j ) + K(e i e 1 ) = (n + 1)a + b i n+1 a + b c = τ(l 3 ) τ(l 3 ) τ(l 3 ) = τ(l 3 ) + 1 (a + b c). In a similar way, by summing the equation 1) with all the equations from 3) to n+1) we have: (vii) τ(l 4 ) + 1 i n+1<j n+1 K(e i e j ) K(e i e ) = na + b 1 i n+1 Also, by summing the remaining equations we have: (viii) τ(l 4 ) + 1 i n+1<j n+1 Then (viii) (vii) implies: K(e i e j ) K(e i e ) = (n + 1)a + c 1 i n+1 a b + c = τ(l 4 ) τ(l 4 ) τ(l 4 ) = τ(l 4 ) + 1 (a b + c).

5 A characterization of generalized quasi-einstein manifolds 93 ) 1). Let p M and {e 1,..., e n, e n+1,..., e n+1 } be an orthonormal frame of T p M such that U = e 1 and V = e. Let L = Sp({e n+,..., e n+1 }) and L 0 = Sp({e,..., e n+1 }). Then L = Sp({e 1,..., e n+1 }) and L 0 = Sp({e 1, e n+,..., e n+1 }). Thus: Ric(e 1 ) = [K(e 1 e ) + K(e 1 e 3 ) K(e 1 e n+1 )] + +[K(e 1 e n+ ) K(e 1 e n+1 )] = [τ(l ) K(e i e j )] + [τ(l 0 ) K(e i e j )] i<j n+1 n+ i<j n+1 = [τ(l) + 1 (a + b + c) τ(l 0)] + [τ(l 0 ) + 1 (a + b c) τ(l)] = a + b. In a similar way one can prove that Ric(e )= a + c and Ric(e i ) = a for all i {3,..., n + 1}. We define now the 1-forms A(X) = g(x, U), B(X) = g(x, V ) such that A(U) = B(V ) = 1 and we consider the following (0, )-tensor P (X, Y ) = ag(x, Y )+ ba(x)a(y )+ cb(x)b(y ). Then Ric(X, X) = P (X, X) for every X Γ(T M). Because the tensors Ric and P are symmetric, it follows that Ric(X, Y ) = P (X, Y ) for every X, Y Γ(T M) and then M is a generalized quasi-einstein manifold. We give the particular version for dimension three. Theorem.. Let M be a Riemannian 3-manifold. Then the following conditions are equivalent: 1) M is a generalized quasi-einstein manifold with Ric(X, Y ) = ag(x, Y ) + ba(x)a(y )+ cb(x)b(y ) for every X, Y Γ(T M), where a, b, c are real scalars and A, B are non-zero 1-forms on M such that A(X) = g(x, U), B(X) = g(x, V ), g(u, V ) = 0 for all vector field X Γ(T M), U, V being unit vector fields. ) a) τ(l 1 ) = 1 (a + b + c), where L 1 = Sp({U, V }), b) τ(l ) = 1 (a + b c), where L is a -plane section orthogonal to V, c) τ(l 3 ) = 1 (a b + c), where L 3 is a -plane section orthogonal to U. Proof. Similar to that of Theorem.1. We can state now the even dimension version of Theorem.1. from above. Theorem.3. Let M be a Riemannian (n)-manifold, n. Then the following conditions are equivalent: 1) M is a generalized quasi-einstein manifold with Ric(X, Y ) = ag(x, Y ) + ba(x)a(y )+ cb(x)b(y ) for every X, Y Γ(T M) where a, b, c are real scalars and A, B are non-zero 1-forms on M such that A(X) = g(x, U), B(X) = g(x, V ), g(u, V ) = 0 for all vector field X Γ(T M), U, V being unit vector fields.

6 94 Dan Dumitru ) a) τ(l 1 ) = τ(l 1 ) + 1 (b + c) for any n-plane section L 1 T p M such that U, V / L 1, b) τ(l ) = τ(l ) 1 (b + c) for any n-plane section L T p M such that U, V L, c) τ(l 3 ) = τ(l 3 ) + 1 (b c) for any n-plane section L 3 T p M such that U / L 1, V L 3, d) τ(l 4 ) = τ(l 4 ) + 1 ( b + c) for any n-plane section L 4 T p M such that U L 4, V / L 4, where L denotes the orthogonal complement of L in T p M for every p M. Proof. Similar to that of Theorem.1. Acknowledgement The author would like to thank the referee for his useful improvements of the paper. References [1] Bejan, C. L., Characterization of quasi-einstein manifolds. An. Stiint. Univ. Al. I. Cuza Iasi Mat. (N.S.) 53 suppl. 1 (007), [] Chaki, M.C., Maity, R.K., On quasi-einstein manifolds. Publ. Math. Debrecen 57 (000), [3] Chen, B. Y., Dillen, F., Verstraelen, L., Vranken, L., Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces. Proc. Am. Math. Soc. 18 No. (000), [4] De, U. C., Ghosh, G.C., On generalized quasi-einstein manifolds. Kyungpook Math. J. 44 (004), [5] Dumitru, D., On Einstein spaces of odd dimensions. Buletinul Universitatii Transilvania, Brasov, 14(49) Seria B, (007), [6] Singer, I. M., Thorpe, J. A., The curvature of 4-dimensional Einstein spaces. Global Analysis, Princeton University Press, (1969), Received by the editors April 17, 011