In these notes of the mini-course given by the author at the XI-th Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 2009 after the introduction of the toric Kähler geometry, we present Calabi’s family of U$$-invariant extremal Kähler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.

The multi-component nonlinear Schrödinger equations related to $$ and $$-type symmetric spaces with non-vanishing boundary conditions are solvable by the inverse scattering method (ISM). We focus our attention on the single threshold case. We formulate the spectral properties of the Lax operator $$, which is the generalized Zakharov-Shabat operator. Next we construct the corresponding fundamental analytic solutions (FAS) and adapt the Wronskian relations for the constant boundary conditions. They allow one to analyze the mapping from the class of allowed potentials $$ to each of the minimal sets of scattering data $$. The ISM for the Lax operator $$ is interpreted as a nonlinear analog of the Fourier-transform method. As appropriate generalizations of the usual exponential functions we use the so-called ‘squared solutions’, which are constructed in terms of the $$ of $$ and the Cartan-Weyl basis of the Lie algebra, relevant to the symmetric space. Finally we derive the completeness relation for the “squared solution” which turns out to provide the map from $$ to each $$. Such decompositions allow one to derive all fundamental properties of the multi-component nonlinear Schrödinger equations.

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We alsoobtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.

We consider the problem of finding exact solutions of Einstein equations describing gravitational fields generated by isolated sources in order to provide theoretical models for astrophysical objects. To this aim, the Newman-Janis Algorithm is described along with the main results obtained through it and some ambiguities arising in dealing with the Newman-Janis Algorithm are reviewed. Many issues related to the introduction of a cosmological constant term are also pointed out and some detailed examples are discussed.

It is known that Seiberg-Witten equations are defined on smooth four dimensional manifolds. In the present work we write down a six dimensional analogue of these equations on $$. To express the first equation, the Dirac equation, we use a unitary representation of complex Clifford algebra $$. For the second equation, a kind of self-duality concept of a two-form is needed, we make use of the decomposition $$. We consider the eight-dimensional part $$ as the space of self-dual two-forms.

An interesting class of axially symmetric surfaces, which generalizes Delaunay’s unduloids and provides solutions of the shape equation is described in explicit parametric form. This class provide the first analytical examples of surfaces with periodic curvatures studied by K. Kenmotsu and leads to some unexpected relationships among Jacobian elliptic functions and their integrals.

In this paper we give a presentation of the basic vacuum relations of Extended Electrodynamics in terms of linear connections.

We determine surfaces of genus zero in four-dimensional hyper-Kähler manifolds whose twistor lifts are harmonic sections. As corollary, we prove that a surface of genus zero in four-dimensional Euclidean space is twistor holomorphic if its twistor lift is harmonic section.

Let $$ be the unit ball and $$ be a lattice of $$. Bearing in mind that all compact Riemann surfaces are discrete quotients of the unit disc $$, Holzapfel conjectures that the discrete ball quotients $$ and their compactifications are widely spread among the smooth projective surfaces. There are known ball quotients $$ of general type, as well as rational, abelian, K3 and elliptic ones. The present note constructs three noncompact ball quotients, which are birational, respectively, to a hyperelliptic, Enriques or a ruled surface with an elliptic base. As a result, we establish that the ball quotient surfaces have representatives in any of the eight Enriques classification classes of smooth projective surfaces.

The $$–spectral sequence is a cohomological theory naturally associated with a space of infinite jets, which allows to write down many concepts of the variational calculus by using the same logic of the standard differential calculus. In this paper we use such a language (called Secondary Calculus by A. Vinogradov) to describe a delicate aspect of the variational calculus: the appearance of some “natural” boundary conditions in the context of variational problems with free boundary (e.g., transversality conditions). We discover that the Euler–Lagrange operator is actually a graded operator, producing simultaneously the standard Euler–Lagrange equations and these new boundary conditions as different homogeneous components of an unique object. Simple applicative examples will be presented.

In this paper we study the second mean curvature for different hypersurfaces in space forms. We furnish some examples and we remind some connections between II-minimality and biharmonicity. The main result consists in proving that there are no II-minimal translation surfaces in the Euclidean three-space.

Here we extend the star products by means of complex symmetric matrices. In this way we obtain a family of star products. Next we consider the star exponentials with respect to these star products, and finally we obtain several interesting identities.

It is shown that the Bohm equations for the phase $$ and squared modulus $$ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum $$ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat grad$$ and $$ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schrödinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.

We develop the Fourier transform interpretation of the inverse scattering method for nonlinear integrable evolution equations associated with a $$ reduced Zakharov-Shabat system for the Lie algebra $$. A simple representative of this integrable hierarchy is the well-known Kaup-Kupershmidt equation. Our results admit a natural extention for nonlinear equations connected to a deeply reduced Zakharov-Shabat system related to an arbitrary simple Lie algebra.

The gravitational interaction of light is analyzed considering its dual characteristic nature, i.e., as an (electromagnetic) wave or as a particle (photon). Considered as an electromagnetic wave, the light can be source 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. It is shown that in the would be quantum theory of gravity they correspond to spin−1 massless particles.

Partial differential Hamiltonian systems have been recently introduced by the author in arXiv:0903.4528. They are field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds. We will present here their main geometry, consisting of partial differential Hamilton equations, partial differential Noether theorem, partial differential Poisson bracket, etc.