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The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root [Formula: see text]-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root [Formula: see text]-metric be Einstein-reversible.

Recurrent curvature properties are considered on four-dimensional pseudo-Riemannian homogeneous manifolds with non-trivial isotropy, and also on some geometric manifolds.

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 four-dimensional homogeneous manifolds satisfying some generalized Einstein conditions.

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools which that deals with the wide class of fractional order dierential equation in the Riemann-Liouville (RL) concept. In the current page, rst, we employ the classical and non-classical Lie symmetries (LS) to acquire similarity reductions of nonlinear fractional far eld Korteweg{de Vries (KdV) equation and second, we nd the related exact solutions for the derived generators. Finally, according to the Lie symmetry generators acquired, we construct conservation laws (cl) for related classical and non-classical vector elds of fractional far eld KdV equation.

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.

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

In this paper, we utilize the Lie symmetry analysis method to calculate new solutions for the Fornberg-Whitham equation (FWE). Applying a reduction method introduced by M. C. Nucci, exact solutions and first integrals of reduced ordinary differential equations (ODEs) are considered. Nonlinear self-adjointness of the FWE is proved and conserved vectors are computed

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

Three-dimensional Lorentzian manifolds with commuting curvature operators are studied. A complete description is given at the algebraic level. Consequences are obtained at the differentiable setting for manifolds which additionally are assumed to be locally symmetric or homogeneous.

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