On Hyperbolic Hsu-Structure Manifold, Recurrent and Symmetry

- In this paper, we have defined recurrence and symmetry of different kinds in H-Hsu-structure manifold. Some theorem establishing relationship between different kinds of recurrent H-HSU-Structure manifold involving equivalent conditions with respect to projective, conformal, conharmonic and concircular curvature tensors has been discussed recurrence parameter have also been studied. Index


1.Introduction
If a differentiable manifold Vn, of differentiability class ∞ .there be in Vn,a vector valued linear function F of class ∞ , satisfying the algebraic equation ̿ = -a r X, for arbitrary vector field X . (1.1) Where ̅ = FX , 0≤ ≤ and 'a'is real or imaginary number,then{F} is said to give to Vn a Hyperbolic HSU-structure defined by the equations(1.1) and the manifold Vn is called a Hyperbolic HSU -structure manifold.Hyperbolic HSU-structure manifold or briefly H-Hsu-structure manifold. If a= ±1 and r =2 ,it is an almost complex structure. If a= ± and r = 2 , it is an almost product structure or a hyperbolic almost complex structure. If a = 0, it an almost tangent structure or almost hyperbolic tangent structure. If a≠0, s It is the hyperbolic -structure.
Let the Hsustructure Vn, be endowed with a Hermitian metric tensor g, such that g( ̅ , ̅ ) -a r g(X,Y) =0, Then{F,g} is said to give Vn a hyperbolic Hsu-structure metric manifold. Agreement (1.1):In what follows and the above, the equations containing X,Y,Z…………,etc. hold for these arbitrary vector in Vn, The curvature tensor K,a vector -valued tri-linear function w.r.t the connexion D is given by Theorem(2.6).
In a recurrent H-HSU-structure manifold, if any two of following conditions hold for the same recurrence parameter, then the third also hold: It is concircular (1) Which shows that the manifolds is concircular -(1)-recurrent. Similarly, it can be shown that if the recurrent manifold is either conformal-(1)-recurrent and concircular-(1)-recurrent or conharmonic-(1)-recurrent and concircular-(1)-recurrent then it is either conharmonic-(1)-recurrent or conformal-(1)-recurrent for same recurrence parameter.
Theorem(2.7) In a (1,2) recurrent H-HSU-structure manifold, if any two of following conditions hold for the same recurrence parameter, then the third also hold: It is concircular (1,2)-recurrent.
In a (1,2,3) recurrent H-HSU-structure manifold, if any two of following conditions hold for the same recurrence parameter, then the third also hold: Where ̅ = FX , 0≤ ≤ and 'a'is real or imaginary number,then{F} is said to give to Vn a Hyperbolic HSU-structure defined by the equations(1.1) and the manifold Vn is called a Hyperbolic HSU -structure manifold.Hyperbolic HSU-structure manifold or briefly H-Hsu-structure manifold. If a= ±1 and r =2 ,it is an almost complex structure. If a= ± and r = 2 , it is an almost product structure or a hyperbolic almost complex structure. If a = 0, it an almost tangent structure or almost hyperbolic tangent structure. If a≠0, s It is the hyperbolic -structure.
Let the Hsustructure Vn, be endowed with a Hermitian metric tensor g, such that g( ̅ , ̅ ) -a r g(X,Y) =0, Then{F,g} is said to give Vn a hyperbolic Hsu-structure metric manifold. Agreement  Theorem(2.6).

Theorem (2.8) )
In a (1,2,3) recurrent H-HSU-structure manifold, if any two of following conditions hold for the same recurrence parameter, then the third also hold: