On the symmetries of three-dimensional generalized symmetric spaces

In this work we consider the three-dimensional generalized symmetric space, equipped with the left-invariant pseudo-Riemannian metric. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations

On the symmetries of Lorentzian four-dimensional generalized symmetric spaces of type C

We consider the four-dimensional generalized symmetric spaces of type C, equipped with a left-invariant Lorentzian metric. We completely describe its affine, homothetic and Killing vector fields. We also obtain a full classification of its Ricci, curvature and matter collineations.

Geometric properties of generalized symmetric spaces

It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.

Isomorphy classes of k-involutions of G2

Isomorphy classes of k-involutions have been studied for their correspondence with symmetric k-varieties, also called generalized symmetric spaces. A symmetric k-variety of a k-group G is defined as Gk/Hk where θ : G → G is an automorphism of order 2 that is defined over k and Gk and Hk are the k-rational points of G and H = Gθ, the fixed point group of θ, respectively. This is a continuation of papers written by A. G. Helminck and collaborators [Involutions of SL (2, k), (n > 2), Acta Appl. Math.90(1–2) (2006) 91–119, Classification of involutions of SO (n; k; b), to appear, On the classification of k-involutions, Adv. Math.153(1) (1988) 1–117, Classification of involutions of SL (2, k), Comm. Algebra30(1) (2002) 193–203] expanding on his combinatorial classification over certain fields. Results have been achieved for groups of type A, B and D. Here we begin a series of papers doing the same for algebraic groups of exceptional type.