Prof. MOINDI STEPHEN KIBET

The P1 - curvature tensor defined from W3 - curvature tensor has been studied in the spacetime of general
relativity. The Bianchi like differential identity is satisfied by P1 - tensor if and only if the Ricci tensor is
of Codazzi type. It is shown that Einstein like field equations can be expressed with the help of the
contracted part of P1 - tensor, which is conserved if the energy momentum tensor is Codazzi type.
Considering P1 -flat space time satisfying Einstein’s field equations with cosmological term, the
existence of Killing vector field ξ is shown if and only if the Lie derivative of the energy-momentum
tensor vanishes with respect to ξ, as well as admitting a conformal Killing vector field is established if
and only if the energy-momentum tensor has the symmetry inheritance property. Finally for a P1 - flat
perfect fluid spacetime satisfying Einstein’s equations with cosmological term, some results are obtained

In this paper, we study ƞ-Ricci solitons on Lorentzian para-Sasakian manifold satisfying
R(ξ,X)•W_5(Y,Z)U=0 and W_5(ξ,X)•R(Y,Z)U=0 conditions.
We prove that on a Lorentzian para-Sasakian manifold (M,ξ,ƞ,g), the Ricci curvature tensor satisfying
any one of the given conditions, the existence of ƞ-Ricci soliton then implies that (M,g) is Einstein
manifold. We also conclude that in these cases, there is no Ricci soliton on M, with the potential vector
field ξ (the killing vector)

In this paper, we study ƞ-Ricci solitons on Lorentzian para-Sasakian manifold satisfying
R(ξ,X)•W_5(Y,Z)U=0 and W_5(ξ,X)•R(Y,Z)U=0 conditions.
We prove that on a Lorentzian para-Sasakian manifold (M,ξ,ƞ,g), the Ricci curvature tensor satisfying
any one of the given conditions, the existence of ƞ-Ricci soliton then implies that (M,g) is Einstein
manifold. We also conclude that in these cases, there is no Ricci soliton on M, with the potential vector
field ξ (the killing vector).

In this Paper η- Ricci solitons are considered on Para- Kenmotsu manifolds satisfying (ξ,.)S.W8 = 0 and
(ξ,.)W8.S = 0. The results of Blaga [1] for W2 have motivated us to use the same conditions on W8. We have
proved that the Para- Kenmotsu manifolds satisfying (ξ,.)W8.S = 0. Are quasi- Einstein Manifolds and
those satisfying (ξ,.)S.W8 = 0, are Einstein Manifolds. At the end of the paper it has been proven that the
para- Kenmotsu manifolds with cyclic Ricci tensor and η− Ricci soliton structure are quasi-Einstein
manifolds.

In this research paper we introduce the operators associated with a frame. That is the Analysis and the Synthesis Operators and their basic properties. The structure of matrix representation of the Synthesis operator is also analysed. This matrix is what most frame constructions in fact focus on. The frame operator which is just the joining together of the analysis and synthesis operators is fundamental for the reconstruction of signals form frame coefficients. We also give a complete characterization of the synthesis matrix in terms of the frame operator.

In this research paper we do an introduction to Hilbert space frames. We also discuss various frames in the Hilbert space. A frame is a generalization of a basis. It is useful, for example, in signal processing. It also allows us to expand Hilbert space vectors in terms of a set of other vectors that satisfy a certain condition. This condition guarantees that any vector in the Hilbert space can be reconstructed in a numerically stable way from its frame coe? cients. Our focus will be on frames in? nite dimensional spaces.

In this paper the geometric properties of W3 -
curvature tensor are studied in K-contact Riemannian
manifold.