Abstract
The object of the present paper is to study a quarter-symmetric nonmetric connection on a -Sasakian manifold. In this paper we consider the concircular curvature tensor and conformal curvature tensor on a -Sasakian manifold with respect to the quarter-symmetric nonmetric connection. Next we consider second-order parallel tensor with respect to the quarter-symmetric non-metric connection. Finally we consider submanifolds of an almost paracontact manifold with respect to a quarter-symmetric non-metric connection.
1. Introduction
In 1975, Golab [1] defined and studied quarter-symmetric connection in a differentiable manifold with affine connection.
A linear connection on an -dimensional Riemannian manifold is called a quarter-symmetric connection [1] if its torsion tensor of the connection satisfies where is a 1 form and is a tensor field.
In particular, if , then the quarter-symmetric connection reduces to a semisymmetric connection [2]. Thus the notion of quarter-symmetric connection generalizes the notion of the semisymmetric connection.
If, moreover, a quarter-symmetric connection satisfies the condition for all , where is the Lie algebra of vector fields of the manifold , then is said to be a quarter-symmetric metric connection; otherwise it is said to be a quarter-symmetric nonmetric connection.
After Golab [1], Rastogi [3, 4] continued the systematic study of quarter-symmetric metric connection.
In 1980, Mishra and Pandey [5] studied quarter-symmetric metric connection in Riemannian, Kaehlerian, and Sasakian manifolds.
In 1982, Yano and Imai [6] studied quarter-symmetric metric connection in Hermitian and Kaehlerian manifolds.
In 1991, Mukhopadhyay et al. [7] studied quarter-symmetric metric connection on a Riemannian manifold with an almost complex structure .
In 1997, Biswas and De [8] studied quarter-symmetric metric connection on a Sasakian manifold. In 2000, Ali and Nivas [9] studied quarter-symmetric connection on submanifolds of a manifold. Also in 2008, Sular et al. [10] studied quarter-symmetric metric connection in a Kenmotsu manifold.
Let be a submanifold of an almost paracontact metric manifold with a positive definite metric . Let the induced metric on also be denoted by . The usual Gauss and Weingarten formulae are given, respectively, by where is the induced Riemannian connection on , is the second fundamental form of the immersion, and and are the tangential and normal parts of . From (1.4) and (1.5) one gets
The submanifold of an almost paracontact manifold is called invariant (resp., anti-invariant) if for each point , (resp. . The submanifold is called totally umbilical if , for all , where is the mean curvature vector defined by , where is an orthonormal basis of . The submanifold is called totally geodesic if for all .
The paper is organized as follows. After recalling the basic properties of -Sasakian manifolds in Section 3, we establish the relation between the Riemannian connection and the quarter-symmetric nonmetric connection. In Section 4, we study the curvature tensor, Ricci tensor, scalar curvature, and the first Bianchi identity with respect to the quarter-symmetric nonmetric connection. Section 5 deals with concircular and conformal curvature tensor on a -Sasakian manifold with respect to the quarter-symmetric nonmetric connection and prove that if in a -Sasakian manifold the concircular curvature tensor is invariant under quarter-symmetric nonmetric connection, then the Ricci tensors are equal with respect to the both connections and also prove if a -Sasakian manifold is conformally flat with respect to the quarter-symmetric nonmetric connection, then the manifold is of quasiconstant curvature with respect to the Levi-Civita connection. In the next section we consider second-order parallel tensor with respect to the quarter-symmetric nonmetric connection. In the last section we consider submanifolds of an almost paracontact manifold with respect to a quarter-symmetric nonmetric connection and prove that on an anti-invariant submanifold of aa almost paracontact manifold with a quarter-symmetric nonmetric connection the induced quarter-symmetric non-connection and the induced Riemannian connection are equivalent. Finally, we prove that a submanifold of a -Sasakian manifold with a quarter-symmetric nonmetric connection is also a -Sasakian manifold with respect to the induced quarter-symmetric nonmetric connection.
2. -Sasakian Manifold
An -dimensional differentiable manifold is said to admit an almost paracontact Riemannian structure , [11] where is a -tensor field, is a vector field, is a 1-form, and is a Riemannian metric on such that for all vector fields . The equation is equivalent to , and then is just the metric dual of , where is the Riemannian metric on . If satisfy the following equations: then is called a para-Sasakian manifold or briefly a -Sasakian manifold, [12, 13]. Especially, a -Sasakian manifold is called a special para-Sasakian manifold or briefly a -Sasakian manifold if admits a 1-form satisfying
It is known that in a -Sasakian manifold the following relation holds: for any vector fields .
Let be an -dimensional Riemannian manifold. Then the concircular curvature tensor and the Weyl conformal curvature tensor are defined by [14] for all , respectively, where is the scalar curvature of , and is the symmetric endomorphism of the tangent space at each point corresponding to the Ricci tensor .
We observe immediately from the definition of the concircular curvature tensor that Riemannian manifolds with vanishing concircular curvature tensor are of constant curvature. Thus one can think of the concircular curvature tensor as a measure of the failure of a Riemannian manifold to be of constant curvature. Also necessary and sufficient condition that a Riemannian manifold be reducible to a Euclidian space by a suitable concircular transformation is that its concircular curvature tensor vanishes. Also conformal curvature tensor plays an important role in differential geometry.
A Riemannian manifold of quasiconstant curvature was given by Chen and Yano [15] as a conformally flat manifold with the curvature tensor of type which satisfies the condition where , are scalars, is a nonzero 1-form defined by , and is a unit vector field.
It can be easily seen that if the curvature tensor is of the form (2.9), then the manifold is conformally flat. If , then it reduces to a manifold of constant curvature.
An -dimensional -Sasakian manifold is said to be -Einstein if the Ricci tensor satisfies where and are smooth function on the manifold. If , then the manifold reduces to an Einstein manifold.
3. Relation between the Riemannian Connection and the Quarter-Symmetric Nonmetric Connection
Let be a linear connection and be a Riemannian connection of a -Sasakian manifold such that where is a tensor of type . For to be a quarter-symmetric connection in , we have [1] where
From (1.2) and (3.3) we get and using (1.2) and (3.4) in (3.2) we obtain Hence a quarter-symmetric connection in a -Sasakian manifold is given by
Conversely, we show that a linear connection on a -Sasakian manifold defined by determines a quarter-symmetric connection.
Using (3.7) the torsion tensor of the connection is given by The above equation shows that the connection is quarter-symmetric [1].
Also we have
In virtue of (3.8) and (3.9) we conclude that is a quarter-symmetric nonmetric connection. Therefore (3.6) is the relation between the Riemannian connection and the quarter-symmetric connection on a -Sasakian manifold.
4. Curvature Tensor of a -Sasakian Manifold with Respect to the Quarter-Symmetric Nonmetric Connection
We define the curvature tensor of a -Sasakian manifold with respect to the quarter-symmetric nonmetric connection by
Using (3.7) we obtain which in view of (2.4) and (2.5) yields
A relation between the curvature tensor of with respect to the quarter-symmetric nonmetric connection and the Riemannian connection is given by the relation (4.3). So from (4.3) and (2.3) we have
Taking inner product of (4.3) with we have where .
From (4.6) we can state the following.
Proposition 4.1. If the manifold is of constant curvature with respect to the Levi-Civita connection, then the manifold is of quasiconstant curvature with respect to the quarter-symmetric nonmetric connection.
Also from (4.6) clearly but From (4.3) it is obvious that
Hence we can state that the curvature tensor with respect to the quarter-symmetric nonmetric connection satisfies first Bianchi identity.
Contracting (4.6) over and , we obtain where and is the Ricci tensors of the connection and , respectively. So in a -Sasakian manifold the Ricci tensor with respect to the quarter-symmetric nonmetric connection is symmetric. Also if is Einstein or -Einstein with respect to the Riemannian connection, then is -Einstein with respect to the quarter-symmetric nonmetric connection.
Again contracting (4.10) we have , where and are the scalar curvature of the connection and , respectively. So we have the following.
Proposition 4.2. For a -Sasakian manifold with the quarter-symmetric metric connection (a)the curvature tensor is given by (4.6),(b)the Ricci tensor is given by (4.10),(c)the first Bianchi identity is given by (4.8),(d), (e)the Ricci tensor is symmetric,(f)if is Einstein or -Einstein with respect to the Riemannian connection, then is -Einstein with respect to the quarter-symmetric nonmetric connection.
5. Concircular and Conformal Curvature Tensor on a -Sasakian Manifold with Respect to the Quarter-Symmetric Nonmetric Connection
We define the concircular curvature tensor and conformal curvature tensor on a -Sasakian manifold with respect to the quarter-symmetric nonmetric connection by for all , respectively, where is the scalar curvature, and is the symmetric endomorphism of the tangent space at each point corresponding to the Ricci tensor with respect to quarter-symmetric nonmetric connection.
Using (2.7) and (4.2), (5.1) reduces to
Now if we consider , then from (5.3) we have Using (5.4) in (4.10) we have
So we can state the following.
Theorem 5.1. If in a -Sasakian manifold the concircular curvature tensor is invariant under quarter-symmetric nonmetric connection, then the Ricci tensors are equal with respect to both the connections.
Let us suppose that , and then we get which in view of (5.3) gives So by the use of (2.6), (5.7) yields From this either or, . Now implies
The converse is trivial.
So we can state the following.
Theorem 5.2. An -dimensional -Sasakian manifold with nonzero scalar curvature satisfies the condition if and only if the manifold is an -Einstein manifold of the form (5.9).
Also using (4.3) and (4.10), (5.2) reduces to
Using (2.8) in (5.10) we obtain which is the relation between conformal curvature tensor with respect to Riemannian connection and with respect to the quarter-symmetric nonmetric connection.
Suppose that the -Sasakian manifold is conformally flat with respect to the quarter-symmetric nonmetric connection, that is, . Now from (5.2) we get
Putting in (5.12) and using (4.6) we obtain
Putting this value in (5.12) we have
From this we obtain the following.
Theorem 5.3. If a -Sasakian manifold is conformally flat with respect to the quarter-symmetric nonmetric connection, then the manifold is of quasiconstant curvature with respect to the Levi-Civita connection.
6. Second-Order Parallel Tensor on -Sasakian Manifold with Respect to the Quarter-Symmetric Nonmetric Connection
Definition 6.1. A tensor of second order is said to be a second-order parallel tensor if where denotes the operator of covariant differentiation with respect to the Riemannian connection.
In [16] De proves that on a -Sasakian manifold a second-order symmetric parallel tensor is a constant multiple of the associated metric tensor. In this section we consider a second-order parallel tensor with respect to the quarter-symmetric nonmetric connection defined as .
Then it follows that for arbitrary vector fields on .
Substitution of in (6.1) which gives us
Putting in the above we get
Differentiating (6.4) covariantly along , we get
From the help of (2.2) and (2.3) we get
Hence we can state the following.
Theorem 6.2. On a -Sasakian manifold there is no nonzero second order parallel tensor with respect to the quarter-symmetric nonmetric connection.
As an immediate corollary we can state the following.
Corollary 6.3. There does not exist a Ricci symmetric -sasakian manifold with respect to the quarter-symmetric nonmetric connection.
7. Submanifolds of an Almost Paracontact Manifold with Respect to a Quarter-Symmetric Nonmetric Connection
We define quarter-symmetric nonmetric connection by (3.7). Now if is the induced connection on submanifold from the connection , then we have where is the second fundamental form of in .
For and , we put
Using (7.2), (1.4), and (3.7) from (7.1) we have Now equating tangential and normal parts, we have
From (7.1) we obtain
From (7.7) the torsion tensor with respect to the induced quarter-symmetric nonmetric connection is given by
Also using (7.7) we have Hence we have the following.
Theorem 7.1. The connection induced on a submanifold of an almost paracontact manifold with a quarter-symmetric nonmetric connection is also a quarter-symmetric nonmetric connection.
From (7.5), it follows that if the submanifold is anti-invariant, that is, , then we have the following.
Corollary 7.2. On an anti-invariant submanifold of an almost paracontact manifold with a quarter-symmetric nonmetric connection the induced quarter-symmetric non-connection and the induced Riemannian connection are equivalent.
Let be an orthogonal basis of , where .
From (7.6), we obtain Since, summing up for and dividing by we obtain that is, the mean curvature of the submanifold with respect to the Riemannian connection coincides with that of with respect to the quarter symmetric nonmetric connection.
From (7.6), we have If is totally umbilical with respect to both the Riemannian connection and the quarter symmetric nonmetric connection, then, with the hep of (7.11), from (7.12) we have So, from (7.12) we get for all , Putting in (7.14) we obtain that , for all , which implies that is an invariant submanifold. The converse is trivial. So we have the following.
Theorem 7.3. If is totally umbilical with respect to both the connections, then is invariant. Conversely, if is invariant, then is totally umbilical (resp., totally geodesic) with respect to quarter-symmetric connection if and only if is totally umbilical (resp., totally geodesic) with respect to the Riemannian connection.
Let us consider that the ambient manifold is a -Sasakian manifold. Using (3.6) we have Therefore we have the following.
Proposition 7.4. If is a -Sasakian manifold admitting a quarter-symmetric nonmetric connection, then is also a P-Sasakian manifold with respect to the quarter-symmetric nonmetric connection.
Also induced quarter-symmetric connection is given by (7.5), and using this relation we have
Therefore we have the following.
Theorem 7.5. A submanifold of a -Sasakian manifold with a quarter-symmetric nonmetric connection is also a -Sasakian manifold with respect to the induced quarter-symmetric nonmetric connection.