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Question: CURVATURE OF GENERALIZED SASAKIAN SPACE FORMS

Given an almost contact metric manifold M(\phi,\xi,\eta, g), we say
that M is a generalized Sasakian-space-form if there exist three functions f1, f2, f3
on M such that the curvature tensor R is given by

R(X,Y)Z=f_1{g(Y,Z)X-g(X,Z)Y}+f_2{g(X,\phiZ)\phiY-g(Y,\phiZ)\phiX+2g(X,\phiY)\phiZ}+f_3{\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X+gg(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi}

In (2n+1) dimensional generalized Sasakian space form M2n+1(f_1,f_2,f_3), we have the following relations.

S(X,Y)=(2nf_1+3f_2-f_3)g(X,Y)-(3f_2+(2n-1)f_3)\eta(X)\eta(Y)

S(X,\xi)=2n(f_1-f_3)\eta(X)

C\bar(\xi,X)Y=[f_1-f_3-(r/2n(2n-1))][g(X,Y)\xi-\eta(Y)X]

P(X,Y)Z=R(X,Y)Z-(1/(n-1))[S(Y,Z)X-S(X,Z)Y]

R(X,Y)\xi=(f_1-f_3){\eta(Y)X-\eta(X)Y}

R(\xi,X)Y=(f_1-f_3){g(X,Y)\xi-\eta(Y)X}

for any vector fields X, Y on M, where R, S, C\bar, and r denote the Riemannian curvature tensor, Ricci tensor, concircular curvature tensor and scalar curvature of M2n+1(f1, f2, f3), respectively 

Using above equations I have to evaluate P(C\bar(\xi,X)Y,Z)U.

Manually It is tedious job. Can we find the value by maple? Is there any option to solve these type of problems?

If yes, I can solve many more, which helps a lot in my work.. Thanks in advance.

 

 

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