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The notion of $Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $g$ of $G$ admits a $Gamma$-grading where $Gamma$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $mathbb{H}_3$ adapted to the symmetries of a $Gamma$-symmetric structure on $mathbb{H}_3$. We prove that the classification of $z_2^2$-symmetric Riemannian and Lorentzian metrics on $mathbb{H}_3$ corresponds to the classification of left invariant Riemannian and Lorentzian metrics, up to isometries. This gives examples of non-symmetric Lorentzian homogeneous spaces.
The notion of $Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $g$ of $G$ admits a $Gamm
The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo
We study nice nilpotent Lie algebras admitting a diagonal nilsoliton metric. We classify nice Riemannian nilsolitons up to dimension $9$. For general signature, we show that determining whether a nilpotent nice Lie algebra admits a nilsoliton metric
We exhibit several families of Jacobi-Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi-Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of $mathbb{Z}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag ma