This paper computes the bases of the image of the 2-adic logarithm on the group of the principal units in all 7 quadratic extensions of Q2. This helps one understand the free module structure of the 2-adic logarithm at arbitrary points on its domain. We discuss some applications at the end.

In this paper, we present global norm of potential vector field in Ricci soliton. In particular, we deduce certain conditions so that the potential vector field has finite global norm in expanding Ricci soliton. In addition, we show that if the potential vector field has finite global norm in complete non-compact Ricci soliton having finite volume, then the scalar curvature becomes constant.

In this paper, we have studied the striction curves of a normal ruled surface. We have shown that the evolute of a base curve is the striction curve of a normal ruled surface and the singularities of such surface lie on the evolute of the base curve. We have proved that the striction curves orthogonally cut the planar base curves. Also, we have proved that the surface area between a planar base curve and striction curve of a normal ruled surface is identical to the surface area between the evolutes of the base curve and striction curve. We have obtained different conditions for which the striction curve coincides with the base curve of some ruled surfaces. We have proved that the striction curve of a tangential Darboux developable of a space curve coincides with the base curve if and only if the space curve is a helix. We have deduced a beautiful form of surface area between the base curve and striction curve of a tangential Darboux developable.

This paper is concerned with the study of generalized gradient Ricci-Yamabe solitons. We characterize the compact generalized gradient Ricci-Yamabe soliton and find certain conditions under which the scalar curvature becomes constant. The estimation of Ricci curvature is deduced and also an isometry theorem is found in gradient Ricci-Yamabe soliton satisfying a finite weighted Dirichlet integral. Further, it is proved that a Ricci-Yamabe soliton reduces to an Einstein manifold when the potential vector field becomes concircular. Moreover, the eigenvalue and the corresponding eigenspace of the Ricci operator are also discussed in case of a Ricci-Yamabe soliton with concircular potential vector field.

The purpose of the article is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson-Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost Ricci soliton, almost $\eta$-Ricci soliton, almost gradient $\eta$-Ricci soliton. As a generalization of curvature inheritance \cite{Duggal1992} and curvature collineation \cite{KLD1969}, in this paper, we introduce the notion of \textit{generalized curvature inheritance} and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp. Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation.