On harmonicity and Miao–Tam critical metrics in a perfect fluid spacetime

Abstract

The curvature properties of a perfect fluid spacetime have been studied when the Riemann curvature, the projective curvature, the concircular curvature, the conformal curvature and the conharmonic curvature tensors are, respectively, harmonic. In the flatness case, if the velocity vector of the fluid is torse-forming, we can completely characterize it. Also, we study the consequences of the existence of g-Einstein, g-Ricci and g-Yamabe solitons in perfect fluid spacetime satisfying Miao-Tam critical metric condition.

Spacetimes admitting the Z-symmetric tensor

“…Every GRW spacetime with n = 4 is a perfect fluid spacetime if and only if it is a RW spacetime. We cite ( [4], [5], [13], [16], [19], [20], [22], [31]) and its references for some deep results of the perfect fluid spacetimes.…”

Ricci solitons in a generalized weakly (Ricci) symmetric D-homothetically deformed Kenmotsu manifold

“…Proof. If V = grad(f ), then 1 2 £ V g = Hess(f ) and equation 36 Replacing now Ric, Hess(f ) and ∆ from (5), (10) and (11) in (53), we obtain:…”

“…Moreover, for any f ∈ C ∞ (M), the Hessian, the gradient, the divergence and the Laplace operators w.r.t. g satisfy: (10) Hess…”

“…becomes (47)Hess(f ) + Ric = λg.Using(5) and(10) in (47) we get(48) Hess(f ) + Ric + a − 1 a [2n − ξ(f )](g − η ⊗ η) = λ[ag + a(a − 1)η ⊗ η]and tracing (48) w.r.t. g, we obtain(49) scal = a(2n + a)λ − ∆(f ) + 2n(a − 1) a ξ(f ) − 4n 2 (a − 1) a .Now tracing (47) w.r.t.…”

Harmonic Aspects in an $\eta$-Ricci Soliton

“…A particular case of solitons arise when they evolve by diffeomorphism generated by a gradient vector field, namely when the potential vector field is the gradient of a smooth function. The gradient vector fields play a central rôle in Morse-Smale theory [37] and some aspects of gradient η-Ricci solitons were discusses by author in [3], [7], [8], [10], [12], [16].…”