We present geometrical aspects of Lie groups and reductive homogeneous spaces, and some resent results on homogeneous geodesics and homogeneous Einstein metrics. The article is based on the four lectures given in Varna, June 2007.

New reductions for the multicomponent modified Korteveg de Vries (MMKdV) equations on the symmetric spaces of DIII-type are derived using the approach based on the reduction group introduced by A. Mikhailov. The relevant inverse scattering problem is studied and reduced to a Riemann- Hilbert problem. The minimal sets of scattering data $\mathcal{T}_{i},i = 1,2$ which allow one to reconstruct uniquely both the scattering matrix and the potential of the Lax operator are defined. The effect of the new reductions on the hierarchy of Hamiltonian structures of MMKdV and on $\mathcal{T}_i$ are studied. We illustrate our results by the MMKdV equations related to the algebra $\mathfrak{g} \simeq \mathfrak {so}(8)$ and derive several new MMKdV-type equations using group of reductions isomorphic to $\mathbb{Z}_{2}, \mathbb{Z}_{3},\mathbb{Z}_{4}$.

Discussed is the problem of the mutual interaction between spinor and gravitational fields. The special stress is laid on the problem of the proper choice of the gauge group responsible for the spinorial geometrodynamics. According to some standard views this is to be the local, i.e., $x$-dependent, group SL$(2,\mathbb{C})$, the covering group of the Lorentz group which rules the internal degrees of freedom of gravitational cotetrad. Our idea is that this group should be replaced by SU$(2,2)$, i.e., the covering group of the Lorentz group in four dimensions. This leads to the idea of Klein-Gordon-Dirac equation which in a slightly different context was discovered by Barut and coworkers. The idea seems to explain the strange phenomenon of appearing leptons and quarks in characteristic, mysterious doublets in the electroweak interaction.

We propose a problem in differential geometry, i.e., which dformed surfaces produce prescribed curvature induced quantum potentials. We solve this inverse problem in the case of surfaces of revolution. We also show that there exist rotational surfaces in the form of a circular strip around the axis of symmetry which allow particles with generic angular momentum to bind. The quantum physics of a collection of circular strips of curved surfaces glued together is discussed in the conclusion in view of the possibility to engineer devices based on thin films.

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics, etc. As geometric objects the orbits were the subject of extensive study. However, they remain hard for calculation and application. We propose a simple solution for the following problem: an explicit parametrization of the orbit by means of a generalized stereographic projection, which provide a Kählerian structure on the orbit, and basis two-forms for the cohomology group of the orbit.

We exhibit an example of a homogeneous Lorentzian manifold $G/H$ whose homogeneous geodesics form just the light-cone and a hyperplane in the tangent space. For all light-like homogeneous geodesics, the natural parameter of the orbit is not the affine parameter of the geodesic.

The consideration of some non-standard parametric Lagrangian leads to a fictitious dynamical system which turns out to be equivalent to the Euler problem for finding out all possible shapes of the lamina. Integrating the respective differential equations one arrives at novel explicit parameterizations of the Euler’s elastica curves. The geometry of the inflexional elastica and especially that of the figure “eight” shape is studied in some detail and the close relationship between the elastica problem and mathematical pendulum is outlined.

The notion of photon-like objects is introduced and briefly discussed. The nonlinear connection view on the Frobenius integrability theory on manifolds is considered as a frame in which appropriate description of photon-like objects to be developed.

We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle of $M$ and $G$ is the general natural lifted metric of $g$, has constant sectional curvature.

We find the Legendre curves and a class of integral surfaces in a 7-dimensional three-Sasakian manifold whose mean curvature vectors are eigenvectors of the Laplacian or the normal Laplacian and we give the ex- plicit expression for such surfaces in the sphere $\mathbb{S}^{7}$.

We present stationary and travelling wave solutions for equations describing Bose-Fermi mixtures in an external potentials which are elliptic functions of modulus $k$. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.

Here we demonstrate the existence of a local Darboux chart for the Manev model such that its dynamics becomes locally equivalent to the Kepler model. This explains lot of similarities between these two models and especially why they share common symmetry algebras. We also discuss the existence of group actions on the phase space for the algebras inherent in the Manev model.

We study a diffeological Calculus for rough loop spaces.

A new variational property of geodesics in (pseudo-)Riemannian and Finsler spaces has been found.

We introduce a new notion of a curvature of a superconnection, different from the one obtained by a purely algebraic analogy with the curvature of a linear connection. The naturalness of this new notion of a curvature of a superconnection comes from the study of the singularities of smooth sections of vector bundles (Catastrophe Theory). We demonstrate that the classical examples of obstructions to a local equivalence: exterior differential for two-forms, Riemannian tensor, Weil tensor, curvature of a linear connection and Nijenhuis tensor can be treated in terms of some general approach. This approach, applied to the superconnection leads to a new notion of a curvature (proposed in the paper) of a superconnection.

A MATHEMATICA$^{®}$ package for finding Lie symmetries of partial differential equations is presented. The package has been applied to perform a full Lie group analysis of basic models of nonlinear fiber optics. As a result of this group invariant solutions have been obtained. Comparisons with earlier published computer algebra implementations of the Lie group method are discussed.

Here we give some necessary and sufficient conditions for the validity of the Saxon-Hutner conjecture concerning the preservation of the energy gaps into an infinite one-dimensional lattice.

This paper presents the formalism of symplectic supermanifolds with a non-homogeneous symplectic form and their prequantization.

We regard a cubic spectral problem associated with the Kaup-Kupershmidt equation. For this spectral problem we prove a completeness of its “squared” solutions and derive the completeness relations which they satisfy. The spectral problem under consideration can be naturally viewed as a $\mathbb{Z}_{3}$-reduced Zakharov-Shabat problem related to the algebra $\mathfrak{sl}(3,\mathbb{C})$. This observation is crucial for our considerations.

Physical properties of gravitational waves, belonging to the larger class of exact solutions of Einstein field equations which are invariant for a non-Abelian two-dimensional Lie algebra of Killing fields, are described. It is shown that in the would be quantum theory of gravity they correspond to spin $−1$ massless particles. The gravitational interaction of two pencils of light is analyzed.