An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac.

In this work we investigate the possible classes of seven-dimensional almost paracontact metric structures induced by the three-forms of $G_2^*$ structures. We write the projections that determine to which class the almost paracontact structure belongs, by using the properties of the $G_2^*$ structures. Then we study the properties that the characteristic vector field of the almost paracontact metric structure should have such that the structure belongs to a specific subclass of almost paracontact metric structures.

In this paper, we give a complete classification of Lagrangian and bi-Lagrangian subalgebras, up to an inner automorphism on $\frak{aff}(2,\mathbb{R})$, and compute the curvatures of some bi-Lagrangian structures.

The concepts for inflatable deployable structures have been under development and evaluation for many years. Strangely enough only the Mylar balloon up to now has been described adequately. Here we provide the mathematical model and its analytical solution for the torus-like balloons. Their characteristics and shapes are described explicitly in terms of elliptic integrals. The obtained results are commented shortly and the possible directions for the related studies in the future are outlined.

The aim of this study is to examine the slant curves in Lorentzian manifolds with BCV (Bianchi-Cartan-Vranceanu) metrics. A practical form of the directional derivative in the presence of the BCV metrics is presented. Moreover, some theorems for slant curves in Lorentzian BCV manifolds are proved.