We consider four-dimensional Lie groups equipped with left-invariant Einstein Lorentzian metrics. The harmonicity properties of left-invariant vector fields on these spaces are determined. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Left-invariant vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.

Let G be a locally compact Hausdorff topological group and H be a compact subgroup of G. Then, the homogeneous space G/H possesses a specific Radon measure, which is called a relatively invariant measure. We show that the concepts of convolution and involution can be extended to the integrable functions defined on this homogeneous space. We study the properties of convolution and prove that the space of integrable functions is an involutive Banach algebra with an approximate identity. We also find a necessary and sufficient condition on a closed subspace of this Banach algebra to make it a left ideal.

In this paper we study hereditarily homogeneous general-ized topological spaces. Various properties of hereditarily homogeneous generalized topological spaces are discussed. We prove that a general-ized topological space is hereditarily homogeneous if and only if every transposition of X is a -homeomorphism on X.

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We study curvature properties of four-dimensional Lorentzian manifolds with two-symmetry property. We then consider Einstein-like metrics, Ricci solitons and homogeneity over these spaces.