On a Hsu-unified Structure Manifold with a Quarter- symmetric Non-metric Connection

Abstract: In the present paper, we have defined a Hsu-unified structure manifold and a Hsu-Kahler manifold and studied some properties of the quarter-symmetric non-metric connection. Certain interesting results on such manifolds have been obtained. We have also studied the properties of the contravariant almost analytic vector field on these manifolds equipped with the quarter-symmetric non-metric connection.


Introduction
Quarter-symmetric linear connection was introduced and studied by S. Golab [1] in 1975.Several properties of quarter-symmetric metric and non-metric connections on a differentiable manifold have been studied by Yano and Imai [10], Sular etal. [8], Sengupta and Biswas [7] and many other geometors. In the present paper, we have studied some properties of the quarter-symmetric nonmetric connection on a manifold called Hsu-unified structure manifold and a Hsu-Kahler manifold which is a particular case of Hsu-unified structure manifold satisfying a certain condition. It has been shown that the Nijenhuis tensor with respect to quarter-symmetric non-metric connection and with respect to Riemannian connectionD coincide in the Hsu-unified structure manifold but in the Hsu-Kahler manifold Nijenhuis tensor with respect to vanishes identically i.e. a Hsu-Kahler manifold is integrable. It has also been proved that a contravariant almost analytic vector fieldV with respect to Riemannian connection D is also contravariant almost analytic with respect to quarter-symmetric non-metric connection in the Hsu-Kahler manifold but in the Hsu-unified structure manifold, it is possible with a specific condition

Preliminaries
If on an even dimensional differentiable manifold , = 2 of differentiability class ∞ , there exists a vector valued real linear function F of differentiability class ∞ , satisfying 2 = (2.1) for arbitrary vector field X . Also there exists a Riemannian metric g , such that g ( ̅ , ̅ ) = g(X,Y) where ̅ = , 0 ≤ ≤ and a is a real or complex number. Then in view of the equations (2.1) and (2.2), M n is said to be a Hsu-unified structure manifold.
Let us define a 2-form 'F in M n , given as  The equation (2.6) shows that the 2-form 'F is symmetric in M n . If the Hsu-unified structure manifold M n satisfies a condition ( Then M n will be said to be a Hsu-K ̈ hler manifold. From the equation (2.7), it can be seen that

Quarter-symmetric Non-metric Connection
A linear connection  defined as [9] (3.1) or arbitrary vector fields X andY , is said to be a quarter-symmetric non-metric connection if the torsion tensorS of the connection  and the metric tensorg are given by

Hsu-unified structure manifold equipped with the Quarter-symmetric non-metric connection
In this section, we have the following theorems Theorem 4.1: For a Hsu-unified structure manifold n M equipped with a quarter-symmetric nonmetric connection  , the following results hold good (4.1)

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Proof: From the equations (3.6) and (2.1), we have (4.2) Clearly (4.1) ) (i) follows from (4.2) ) (i and (4.2) ) (ii . InterchangingX andY in (3.6), we get Now the result of (4.1) ) (ii) follows from (4.2) ) (i) and (4.4). Again barringX andY in (3.6) and using (2.1), we obtain By virtue of the equations (3.6) and (4.5), the result of (4.1) ) (iii) follows.   The Nijenhuis tensor with respect to F is a vector valued bilinear function, defined as [2], [3] In view of the equation (2.1), the above expression can be written in the form  Where N*(X,Y) denotes the Nijenhuis tensor with respect to the quarter-symmetric non-metric connection  . Hence we have the statement of the theorem.

Hsu-Kahler manifold with the Quarter-symmetric non-metric connection
As discussed earlier that if the Hsu-unified structure manifold M n satisfies the condition (2.7), then M n is called a Hsu-Kahler manifold. In this section, we have some following theorems  Interchanging X andY in (5.14), we have In view of the equations (5.12), (5.13), (5.14) and (5.15), the equation (5.11) takes the form which proves the theorem.

Contravariant almost analytic vector field
A vector field V is said to be contravariant almost analytic if the Lie-derivative of the tensor field F with respect to V vanishes identically i.e. [2], [3] In a Hsu-unified structure manifold, the equation (6.2) takes the form Now we have the following theorems

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Operating both sides of the equation (6.5) by F , we get As we know that In consequence with the equation (6.9), the last expression becomes Since V is contravariant almost analytic vector field with respect to the Riemannian connection D , so using (6.3), the equation (6.11) is transformed into If the vector field V is also contravariant almost analytic with respect to the quarter-symmetric nonmetric connection , then it will satisfy Hence the equation (6.12) proves the statement of the theorem.

Theorem 6.2:
On a Hsu-Kahler manifold, a contravariant almost analytic vector field V with respect to the Riemannian connectionD is also contravariant almost analytic with respect to the quarter-symmetric non-metric connection.
Proof: Subtraction of the equation (6.7) from (6.6) gives Since V is contravariant almost analytic vector field with respect to the Riemannian connection D in the Hsu-Kahler manifold, so with the help of the equation (6.4), (6.13) yields which implies that V is also contravariant almost analytic vector field with respect to the quartersymmetric non-metric connection .