Three-dimensional Lorentzian manifolds whose skew-symmetric curvature operators have constant eigenvalues are studied. A complete algebraic description is given, which allows a complete characterization at the differentiable level of manifolds which additionally are assumed to be locally symmetric or homogeneous.

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 formM = K × f1 M 1 × f2 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 equationsḠ AB = −Λḡ AB , with cosmological constantΛ, which are reducible to the Einstein equations G αβ = −Λ 1 g αβ and G ij = −Λ 2 h ij on the submanifolds M 1 , M 2 , with cosmological constants Λ 1 and Λ 2 , respectively, whereΛ, Λ 1 and Λ 2 are functions of f 1 , f 2 and n 1 , n 2 .

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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 (M ,ḡ) and classify all such spaces satisfying Einstein equationsḠ = −Λḡ. We show that the warping function not only can determine the cosmological constantΛ but also it can determine the cosmological constant Λ appearing in the induced Einstein equations G = −Λh on (M 2 , h). Moreover, we discuss on the origin of the 4D cosmological constant as an emergent effect of higher dimensional warped spaces.

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.

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