This is an overview of classical and recent results on the geometry of isothermic surfaces and conformally flat hypersurfaces in conformal geometry and their relation with curved flats, a particularly simple type of integrable system

Here we discuss the concept of essential nonlinearity, i.e., one which cannot be meaningfully decomposed into well-defined linear background and “small” nonlinear correction. Therefore, the traditional perturbative techniques and asymptotic methods are non-effective then. Two well-established classes of essentially nonlinear field theories exist in the market: 1) The General Relativity and other generally covariant schemes, 2) The Born–Infeld type nonlinearity in traditional and generalized sense. The essential nonlinearity of 1) is intimately connected with the invariance under the very huge group Diff($M$) of all space–time diffeomorphisms. The Born–Infeld scheme 2) is also geometrically motivated by the theory of scalar densities in manifolds. But there is no explicitly seen relationship between these two types of nonlinearities. Below we show that there exists however some hidden link between general covariance and Born–Infeld mechanism. The structure of the group of internal symmetries (target space symmetries) is also relevant. Roughly speaking, “huge” symmetry groups are intimately connected with essential strong nonlinearities. It is so even in finite-dimensional analytical mechanics. Let us remind our affinely-invariant models in mechanics of homogeneously deformable bodies [18–21,23,25–29]. There is no systematic theory, nevertheless some rough although convincing arguments do exist. This essay is just concentrated around the study of the interplay between (high) symmetries and (essential) nonlinearities. The examples quoted below confirm the idea and exhibit a kinship between general covariance and Born–Infeld paradigm. The special stress is laid on models which, by abuse of language, resemble the structured continua with affine geometry of degrees of freedom. These models are based on the bundles $\mathrm{L}M = T^{1}_{1}M$ over the “space–time” manifold $M$, and $F M$, the principal bundle of linear frames. And the special stress is laid on scalar multiplets (trivial bundles over $M)$.

In this note we adapt Axler and Ramey’s method of constructing the harmonic part of a homogeneous polynomial to the Fischer decomposition associated to Dirac operators acting on polynomial spinors. The result yields a constructive solution to a Dirichlet-like problem with polynomial boundary data.

Acting within the framework of the geometric quantum mechanics, we give a Jacobi field interpretation of the quantum mechanical uncertainty. Then a link with elliptic curves via the classical integrability of Schrödinger dynamics and the cross ratio interpretation of quantum probabilities is established. Furthermore a geometrical construction of all special unitary representations of the three-strand braid group on the quantum one-qubit space is provided.

Let $X$ be a closed and oriented Riemannian four-manifold with $b^{+}_2 (X) \gt 1$. We discuss the Seiberg–Witten invariants of $X$ and finite group actions on spin$^c$ structures of $X$. We introduce and comment some of our results on the subject.

We modify the metrics on six-dimensional and seven-dimensional Riemannian g.o. manifolds constructed in previous published papers and we obtain pseudo-Riemannian g.o. manifolds. We describe geodesic graphs of corresponding g.o. spaces. We show that if these geodesic graphs are nonlinear, they are discontinuous on an nonempty set but they are continuous at the origin.

In this paper we establish some existence and uniqueness theorems for a Plateau problem at infinity for complete spacelike surfaces in $S^{3}_{1}$ whose mean and Gauss–Kronecker curvatures verify the linear relationship $2 \varepsilon (H − 1) − (\varepsilon + 1)(K − 1) = 0$ for $− \varepsilon \in \mathbb{R}^+$.

In this paper we introduce the notion of a semi-developable surface of codimension two as a generalization of the notion of a developable surface of codimension two. We give a characterization of the developable and semi-developable surfaces in terms of their second fundamental forms. We prove that any hypersurface of conullity two in Euclidean space is locally a foliation of developable or semi-developable surfaces of codimension two.

We consider $N$-wave type equations related to symplectic and orthogonal algebras. We obtain their soliton solutions in the case when two different $\mathbb{Z}_2$ reductions (or equivalently one $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$-reduction) are imposed. For that purpose we apply a particular case of an auto-Bäcklund transformation – the Zakharov–Shabat dressing method. The corresponding dressing factor is consistent with the $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$-reduction. These soliton solutions represent $N$-wave breather-like solitons. The discrete eigenvalues of the Lax operators connected with these solitons form “quadruplets” of points which are symmetrically situated with respect to the coordinate axes.

The localized induction equation for the stretched vortex filament is reviewed. The transport of the momentum and the angular momentum carried by the stretching vortex filament is studied. The generalizations of the equation are considered.

We describe a geometrical interpretation of Topological Quantum Mechanics (TQM). Basics of the general topological theories are briefly discussed as well. The appropriate correspondence between objects of TQM and the algebraic topology is pointed out. It is proved that the correlators in TQM can be expressed via intersection numbers of some submanifolds of the target space with paths of steepest descent between critical points. Another correspondence is only conjectured, namely the correspondence between correlators and an integral of Massey products on cohomology classes of the target manifold.

The Manev model and its real form dynamics are known to possess Ermanno–Bernoulli type invariants similar to the Laplace–Runge–Lenz vector of the Kepler model. Using these additional invariants, we demon- strate here that both Manev model and its real Hamiltonian form posses the same $\mathfrak{so}(3)$ or $\mathfrak{so}(2,1)$ symmetry algebras (in addition to the angular momentum algebra) on angular momentum level sets. Thus Kepler and Manev models are shown to have identical symmetry algebras and hence sharing more features than previously expected.

We consider the energy and bienergy functionals as variational problems on the set of Riemannian metrics and present a study of the bi- harmonic stress-energy tensor. This approach is then applied to characterize weak conformality of the Gauss map of a submanifold. Finally, working at the level of functionals, we recover a result of Weiner linking Willmore surfaces and pseudo-umbilicity.

Elsewhere we gave a parametrization of the Mylar balloon in terms of elliptic functions. Here, we parametrize other linear Weingarten surfaces using the incomplete Beta function and the hypergeometric function. We also provide a corresponding variational characterization and Maple procedures which plot the linear Weingarten surfaces.

In this paper we study a Riemanian metric on the tangent bundle $T(M)$ of a Riemannian manifold $M$ which generalizes Sasakian metric and Cheeger–Gromoll metric along a compatible almost complex structure which together with the metric confers to $T(M)$ a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on $T(M)$. We found conditions under which $T(M)$ is almost Kählerian, locally conformal Kählerian or Kählerian or when $T(M)$ has constant sectional curvature or constant scalar curvature.

We review the theory of Chaplygin gas. This theory arises from the non-relativistic fluid mechanics with exotic state equation. This condition actually admits that the fluid theory reduces to the simple relativistic geometrical objects. This means that the non-relativistic fluid theory has hidden Poincare symmetry. We will show that the geometrical object is the brane described by the Nambu–Goto action. The application of this Chaplygin gas to the universe model is also briefly reviewed.

This paper is an introduction to the theory of algebras with polynomial identities. It stresses on matrix algebras and polynomial identities for them. The notion of Bergman polynomials is introduced. Such types of polynomials are investigated being identities for algebras with symplectic involution. In the Lie case more information is given for Bergman polynomials as Lie identities for the considered algebras.

Here we combine the ideas of a quantum functional theory intended to describe intrinsically extended particles with those of a geometro-stochastic one describing stochastically extended particles. The main ingredients of the former are a physical wave $u$ replacing the point ${\bf x}$ and a functional wave $X[u,t]$ replacing the probability wave function of the conventional quantum theory. The latter introduces a proper wave function accounting for the unavoidable errors in the measurement of continuous observable such as the position and momentum.

The membrane shape equation derived by Helfrich and Ou-Yang describes the equilibrium shapes of biomembranes, built by bilayers of amphiphilic molecules, in terms of the mean and Gaussian curvatures of their middle-surfaces. Here, we present a new class of translationally-invariant solutions to this equation in terms of the elliptic functions which completes the solutions found earlier. In this way, all translationally-invariant solutions to the membrane shape equation are determined. Special attention is paid to those translationally-invariant solutions of the membrane shape equation which determine closed cylindrical (tube-like) surfaces (membrane shapes). Several examples of such surfaces are presented.

The Conte–Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of our approach to the quintic complex one-dimensional Ginzburg–Landau equation (CGLE5) allows us to find elliptic solutions in the wave form. We also find restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions to the CGLE5.