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In this paper, we give a full classification of pseudo-Riemannian Lie groups of dimension four with non-trivial recurrent curvature tensor (i.e., non-locally symmetric). Then we investigate Ricci solitons on recurrent curvature Lie groups. Locally conformally flat examples of our classification have also been presented.

Special multiply warped product manifold as the generalization of a singly warped product manifold which plays an important role in the geometry and physics were considered. Essentially geodesics of this multiply warped product manifold have been presented.

This work provides an analytical investigation of the time-fractional Kundu–Eckhaus (KE) equation. We use the symmetry of the Lie group as an appropriate tool which deals with the wide class of fractional-order differential equations in Riemann–Liouville sense. In the current work, firstly, we employ classical Lie symmetries to obtain similarity reductions of nonlinear generalised time-fractional KE equation. At the final step, we find relevant exact solutions for the extracted generators.

In this paper, we investigate the Einstein equations with cosmological constant for Randall-Sundrum (RS) and Dvali-Gabadadze-Porrati (DGP) models to determine the warp functions in the context of warp product spacetimes. In RS model, it is shown that Einstein's equation in the bulk is reduced into the brane as a vacuum equation, having vacuum solution, which is not affected by the cosmological constant in the bulk. In DGP model, it is shown that the Einstein's equation in the bulk is reduced into the brane and along the extra dimension, where both equations are affected by the cosmological constant in the bulk. We have solved these equations in DGP model, subject to vanishing cosmological constants on the brane and along extra dimension, and obtained exact solutions for the warp functions. The solutions depend on the typical values of cosmological constant in the bulk as well as the dimension of the brane. So, corresponding to the typical values, some solutions have exponential behaviours which may be set to represent warp inflation on the brane, and some other solutions have oscillating behaviours which may be set to represent warp waves or branes waves along the extra dimension., Comment: 7 pages

We classify left-invariant Einstein-like metrics of neutral signature, over four-dimensional Lie groups. Several geometric properties such as being conformally flat, existing Ricci solitons and Walker structures are exhibited.

It is shown that locally conformally flat weakly-Einstein manifolds are either locally symmetric, and hence a product $$N_1^m(c)\times N_2^m(-c)$$ , or otherwise they are locally homothetic to some specific warped product metrics $${\mathcal {I}}\times _fN(c)$$ . As an application we classify weakly-Einstein hypersurfaces in the Euclidean space.

In this paper, the Painleve analysis is applied to test the integrability of the generalized Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation with time dependent coefficients. Symmetry reductions and some corresponding invariant solutions in the integrable cases are completely considered. Soliton solutions of constant variables case in two integrable cases are reported.

In this paper, warped product in Hermitian and Kählerian manifolds was considered, and almost complex structure on the warped product is defined. Also, warped product almost Hermitian manifolds were defined. According to the holomorphic sectional curvature of Kählerian manifolds, the Einstein condition for constructed warped product Kählerian manifolds is obtained.

A full classification of invariant Einstein-like metrics on four-dimensional pseudo-Riemannian homogeneous spaces with non-trivial isotropy is given. Specially, proper examples of Codazzi manifolds which are not conformally flat have been presented.

We classify all warped product space-times in three categories as i) generalized twisted product structures, ii) base conformal warped product structures and iii) generalized static space-times and then we obtain the Einstein equations with the corresponding cosmological constant by which we can determine uniquely the warp functions in these warped product space-times., 10 pages

Ricci almost solitons on three-dimensional Lorentzian manifolds with recurrent curvature are classified. As an application, we construct homogeneous examples without Riemannian counterpart.

We study Einstein's equation in $(m+n)D$ and $(1+n)D$ warped spaces $(\bar{M},\bar{g})$ and classify all such spaces satisfying Einstein equations $\bar{G}=-\bar{\Lambda}\bar{g}$. We show that the warping function not only can determine the cosmological constant $\bar{\Lambda}$ but also it can determine the cosmological constant ${\Lambda}$ appearing in the induced Einstein equations ${G}=-{\Lambda}{h}$ on $(M_2, h)$. Moreover, we discuss on the origin of the $4D$ cosmological constant as an emergent effect of higher dimensional warped spaces., Comment: 10 pages

It is shown that for every multidimensional metric in the multiply warped product form $\bar{M} = K\times_{f_1} M_1\times_{f_2}M_2$ with warp functions $f_1$, $f_2$, associated to the submanifolds $M_1$, $M_2$ of dimensions $n_1$, $n_2$ respectively, one can find the corresponding Einstein equations $\bar{G}_{AB}=-\bar{\Lambda}\bar{g}_{AB}$, with cosmological constant $\bar{\Lambda}$, which are reducible to the Einstein equations $G_{\alpha\beta} = -\Lambda_1 g_{\alpha\beta}$ and $G_{ij} =-\Lambda_2 h_{ij}$ on the submanifolds $M_1$, $M_2$, with cosmological constants ${\Lambda_1}$ and ${\Lambda_2}$, respectively, where $\bar{\Lambda}$, ${\Lambda_1}$ and ${\Lambda_2}$ are functions of ${f_1}$, ${f_2}$ and $n_1$, $n_2$., Comment: 12 pages, Reference updated

In this paper, we study three-dimensional homogeneous paracontact metric manifolds for which the Reeb vector field of the underlying paracontact structure satisfies a nullity condition. We give example of paraSasakian and non-paraSasakian [Formula: see text]-manifolds. Finally, we exhibit explicit example of [Formula: see text]-Einstein manifolds.