Some Connections in Almost Hermitian Manifold

The idea of this research is to study different types of connections in an almost Hermite manifold. The connection has been established between linear connection and Riemannian connection. Three new linear connections 1 2 3 , , ∇ ∇ ∇ are introduced. The necessary and sufficient condition for 1 2 3 , , ∇ ∇ ∇ to be metric is discussed. A new metric ( ) * , s X Y has been defined for ( ) * , , n M F g and additional properties are discussed. It is also proved that for the quarter symmetric connection ∇ is unique in given manifold. The hessian operator with respect to all connections defined above has also been discussed.


Introduction
The study of connection has been the field of interest for most of the mathematicians. The study of connections, semi symmetric connection was done in detail by Yano [1] followed by Konar and Chaki [2], De and Biswas [3], Pandey and Dubey [4], Pandey and Chaturvedi [5], Andonie [6] and many more, Quarter symmetric connection by Golab [7], Rastogi [8], Mishra and Pandey [9] [10], Biswas and De [11], De and Sengupta [12]. Quarter symmetric non-metric connection was studied in Riemannian, Kaehlerian and Sasakian manifolds. Quarter symmetric non-metric connection was studied in detail by Bhowmik [13], Mondal and De [14], Haseeb, Prakash and Siddiqi [15]. Kankarej [16] has studied the quarter symmetric non-metric connection in almost Hermitian Manifold. In this research three new types of connection have been discussed in al-most Hermitian Manifold and the necessary and sufficient condition for it to be a metric has been discussed. A new metric has been defined and some additional properties with respect to the new metric is discussed.
It is always possible to introduce a Hermite metric on an almost complex ma- nifold.
An almost complex manifold n M endowed with an almost complex structure F and a Hermite metric g is called an almost Hermite manifold with struc- where Φ is a tensor field of type ( ) 1,1 and ω is a 1-form associated with vector field ρ ( ) ( ) If the quarter symmetric connection D satisfies: then the connection D is said to be quarter symmetric metric connection, otherwise it is said to be a quarter symmetric non-metric connection.
A necessary and sufficient condition that vector field X on a Riemannian Ma- for any vector fields Y and Z. The connection D is unique in Riemannian manifold and is also called Levi-Civita connection on n M .

Relation between the Riemannian Connection and a Linear Connection
The set of connections in n M defines a unique (2,1) tensor B such that the tensor B is a subject to the requirement and ∇ is any linear connection The torsion tensor of ∇ is where Φ is a tensor field of type ( ) 1,1 and ω is a differential 1-form.
Theorem 2.1. ω being a differential 1-form is De Rham closed.
As ω is a differential 1-form it can be represented as ω is closed and also it is De Rham closed.

Some Connections on an Almost Hermite Manifold
where is a 1-form associated with vector field ρ and X and Y are vector fields.
Then 1 ∇ is a symmetric connection.
Proof: From (2.1) and (3.1) we have Interchanging X and Y we have Thus it is proved that 1 ∇ is a symmetric connection.
Theorem 3.2. The necessary and sufficient condition for 1 ∇ to be a metric connection is Above equation proves that the given connection 1 ∇ is non metric.
Necessary and sufficient condition for connection 1 ∇ to be metric: Then 2 ∇ is a semi symmetric connection.
From (2.1) and (3.7) we have Interchanging X and Y we have From (3.8) and (3.9) Theorem 3.4. The necessary and sufficient condition for 2 ∇ to be a metric connection is It proves that 2 ∇ defined in (3.7) is non metric.
Necessary and sufficient condition for 2 ∇ to be metric: Then 3 ∇ is a quarter symmetric connection.

From (2.1) and (3.12) we have
Interchanging X and Y we have From (3.13) and (3.14) Theorem 3.6. The necessary and sufficient condition for 3 ∇ to be a metric connection is By the property of φ we have Proof: Let us define a metric g  in the Hermitian metric such that: , , , , , By the property of φ we have ( )