In this paper, we investigate the reflections in Minkowski three-space by three different approaches. Firstly, we define Lorentzian reflections with Lorentzian inner product. Then, we examine Lorentzian reflections in view of Lorentzian Householder matrices. Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we find the matrix representation of Lorentzian reflections and characterize the plane of reflection by using this matrix representation. Moreover, we prove that any Lorentzian orthogonal transformation can be represented by the composition of at most six reflections.

We have derived a new system of mKdV-type equations which can be related to the affine Lie algebra $A^{(2)}_{5}$. This system of partial differential equations is integrable via the inverse scattering method. It admits a Hamiltonian formulation and the corresponding Hamiltonian is also given. The Riemann-Hilbert problem for the Lax operator is formulated and its spectral properties are discussed.

We achieve compositions rules for the geometric parameters of the composed rotations, which is in a certain sense analogous to the well known Rodrigues formula. We also obtain a necessary and sufficient condition for a composition of two simple rotations in $\mathbb{E}^4$ to be a simple rotation.

We study the properties of the modified Riemann extensions evolving under the Ricci flow with examples.

Guided by the Jacobi’s work published one year before his death about the rotation of a rigid body, the behavior of the rotation matrix describing the dynamics of the free rigid body is studied. To illustrate this dynamics one draws on a unit sphere the trace of the three unit vectors, in the body system along the principal directions of inertia. A minimal set of properties of Jacobi’s elliptic functions are used, those which allow to compute with the necessary precision the dynamics of the rigid body without torques, the so called Euler’s top. Emphasis is on the paper published by Jacobi in 1850 on the explicit expression for the components of the rotation matrix. The tool used to compute the trajectories to be drawn are the Jacobi’s Fourier series for theta and eta functions with extremely fast convergence. The Jacobi’s sn, cn and dn functions, which are better known, are used also as ratios of theta functions which permit quick and accurate computation. Finally the main periodic part of the herpolhode curve was computed and graphically represented.

We find the existence conditions of unduloidal and nodoidal menisci between two solid spheres and study their stability under axisymmetric perturbations in the framework of non-spectral theory of stability of axisymmetric menisci between two axisymmetric solid bodies in the absence of gravity.