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