Two reduced standard $k$-symplectic Poisson manifolds with respect to the action of a Lie group $G$ are considered, and the relation between the corresponding Poisson structures is established.

It is shown that in five dimensional Einstein-Gauss-Bonnet theories exact vacuum solutions with non-vanishing torsion can be constructed. Some possible observational effects related to neutrino oscillations are pointed out. In the theory of continuum media (in which suitable defects can be described by localized non vanishing torsion) “the gravitational intuition” is a rather useful tool to describe the physical effects of such defects. A possible astrophysical application is shortly described.

In the first part, the theory of bounded symmetric domains is presented along two main approaches: as special cases of Riemannian symmetric spaces of the noncompact type on one hand, as unit balls in positive Hermitian Jordan triple systems on the other hand. In the second part, an invariant for triples in the Shilov boundary of such a domain is constructed. It generalizes an invariant constructed by E. Cartan for the unit sphere in $\mathbb{C}^2$ and also the triple Maslov index on the Lagrangian manifold.

We summarize a proof for the existence and uniqueness of the loxodrome joining two distinct points $p_0$ and $p_1$ on an open half of an ellipsoid of revolution. We also compute the unique angle $\alpha \in [0,2\pi)$ which the loxodrome makes with the meridians intersecting the loxodrome.