Here we present a review of the results in the theory of the submanifolds related to the notion of area and mean curvature vector. It is shown that if $r$ is the radius vector of the submanifold $F^n$ in the Euclidean space $E^{n+p}, \rho = \frac{1}{2}r^2$ and $H$ is the corresponding mean curvature vector, then the Laplace-Beltrami operator satisfies $\nabla_2\rho = n(1+ rH)$ and therefore in the case of a minimal submanifold one has $\nabla_2\rho = n$. When $n = 2$ it is proven that if we have a complete submanifold $F^n \subset E^n$ with $|H| \leq H_0 = \text{const}$ and Ricci curvature Ric$(\tau) \geq -a^2 = \text{const}$ in a ball of radius $R$ then $ R \geq \frac{n}{2\sqrt{n-2a} + H_0n}$. The Chern’s problem of existence of a complete and unbounded minimal surface is considered in some depth. Next we consider the conditions of stability of the minimal submanifolds. For submanifolds in Riemannian spaces the problem of stability is connected with the Hopf problem about the existence of metrics with positive curvatures on $S^2 \times S^2$. It is shown that a two-dimensional minimal surface $F^2$ homeomorphic to the sphere $S^2$ in a oriented simply-connected manifold $M^n$ with a sectional curvature $\frac{1}{4} \lt K_a \leq 1$ is non-stable. Pogorelov’s result about minimal surfaces in $S^2 \times S^n, n \gt 2$ is outlined as well. Finally, an expression is derived for the Riemannian tensor of a submanifold defined implicitly by a system of equations in the Euclidean space and a remarkably simple formula is found for the Gaussian curvature of $F^2 \subset E^4$.

The purpose of this series of lectures is to provide an informal survey of the background in Donaldson theory required to understand the motivation behind Witten’s construction (in 1988) of the first topological quantum field theory.

In this paper we will report on our recent studies on curvature properties of space-like hypersurfaces in de Sitter space. In particular, we will state certain estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of complete space-like hypersurfaces in de Sitter space. We will also establish a sufficient condition for a compact space-like hypersurface in de Sitter space to be spherical in terms of a lower bound for the square of its mean curvature.

Vesicle shapes with axial symmetry are modeled using the Cassinian oval, which when rotated leads to a variety of surfaces specified as the level sets of algebraic functions. A continuous set of shapes including the sphere, torus, biconcave discocyte and dumbbell are described by considering a Cassinian oval with a focus distance that is a purely imaginary number. Although these surfaces are not exact solutions of the vesicle shape equation, they reveal some qualitative geometric properties that are difficult to examine by currently available numerical or exact solutions. The relation between the volume of the vesicle at constant surface area and the homogeneity index is derived. Going from the implicit to the explicit coordinatization of these surfaces, we find their fundamental forms and curvatures in a form convenient for experimenters. The integrals specifying the volume and surface area are evaluated analytically in a form relevant to the sphere geometry. The free energy corresponding to the bending of the outer membrane of the vesicles, modeled by the Cassinian oval, is plotted numerically as a function of the form factor or the reduced volume. The effect of the so-called Helfrich spontaneous curvature on the energy minimization is also examined

In [1], the generalization of Laguerre’s function of direction for a surface in ordinary space to a hypersurface of a Riemannian space is obtained. The Laguerre’s function of direction for a hypersurface of a Weyl space has been derived in [2]. In this paper, the generalization of Laguerre’s function of direction to a hypersurface of generalized Weyl space is made.

The symplectic leaves of W-algebras are the intersections of the symplectic leaves of the Kac-Moody algebras and the hypersurface of the second class constraints, which define the $W-$algebra. This viewpoint enables us to classify the symplectic leaves and also to give a representative for each of them. The case of the $W_2$ (Virasoro) algebra is investigated in detail, where the positivity of the energy functional is also analyzed.

Quantum holography is a well established theory of mathematical physics based on harmonic analysis on the Heisenberg Lie group G. The geometric quantization is performed by projectivization of the complexified coadjoint orbit picture of the unitary dual $\hat{G}$ of $G$ in order to achieve a geometric adjustment of the quantum scenario to special relativity theory. It admits applications to various imaging modalities such as synthetic aperture radar (SAR) in the microwave range, and, most importantly for the field of non-invasive medical diagnosis, to the clinical imaging modality of magnetic resonance tomography (MRI) in the radiofrequency range. Quantum holography explains the quantum teleportation phenomenon through Einstein-Podolsky-Rosen (EPR) channels which is a consequence of the non-locality of phase coherent quantum field theory, the concept of absolute simultaneity of special relativity theory which provides the Einstein equivalence of energy and Fitzgerald-Lorentz dilated mass, and the perfect holographic replication process of molecular genetics. It specifically reveals what was before unobservable in quantum optics, namely the interference phenomena of matter wavelets of Bose-Einstein condensates, and what was before unobservable in special relativity, namely the light in flight (LIF) recording processing by ultrafast laser pulse trains. Finally, it provides a Lie group theoretical approach to the Kruskal coordinatized Schwarzschild manifold of quantum cosmology with large scale applications to general relativity theory such as gravitational instanton symmetries and the theory of black holes. The direct spinorial detection of gravitational wavelets emitted by the binary radio pulsar PSR 1913+16 and known only indirectly so far will also be based on the principles of quantum holography applied to very large array (VLA) radio interferometers.

In this paper, we show that the first Bianchi identity is valid for a generalized Weyl space having a semi-symmetric $E$-connection and that the second Bianchi identity is satisfied for a recurrent generalized Weyl space provided that the recurrence vector $\psi_l$ and the Vranceanu vector $\Omega_l$ are related by $\psi_l = \frac{2}{n-1} \Omega_l$.

The reductions of the integrable $N$-wave type equations solvable by the inverse scattering method with the generalized Zakharov–Shabat system $L$ and related to some simple Lie algebra $\mathfrak{g}$ are analyzed. Special attention is paid to the $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$-reductions including ones that can be embedded also in the Weyl group of $\mathfrak{g}$. The consequences of these restrictions on the properties of their Hamiltonian structures are analyzed on specific examples which find applications to nonlinear optics.

We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo-Riemannian 3-space. We define two dimensional surfaces conformally related to the minimal surfaces in flat three dimensional geometries which enable us to give a construction of the metrics of some even dimensional Ricci flat (pseudo-) Riemannian geometries.

In these notes, we will discuss from a purely geometric point of view classical mechanics on certain type of Grassmannians and discs. We will briefly discuss a superversion which in some sense combines these two models, and corresponds to the large-$N_c$ limit of $SU(N_c)$ gauge theory with fermionic and bosonic matter fields, both in the fundamental representation, in $1 + 1$ dimensions [12]. This result is a natural extension of ideas in [16]. There it has been shown that the large-$N_c$ phase space of $1+1$ dimensional QCD is given by an infinite dimensional Grassmannian. The complex scalar field version of this theory is worked out in [18] and it is shown that the phase space is an infinite dimensional disc.

We consider the variety of $(p+1)$-tuples of matrices $M_j$ from given conjugacy classes $C_j \subset \text{GL}(n,\mathbb{C})$ such that $M_1 \cdot M_{p+1}=I$. This variety is connected with the Deligne-Simpson problem: Give necessary and sufficient conditions on the choice of the conjugacy classes $C_j \subset \text{GL}(n,\mathbb{C})$ such that there exist irreducible $(p+1)$-tuples of matrices $M_j \in C_j$ whose product equals $I$. The matrices $M_j$ are interpreted as monodromy operators of regular linear systems on Riemann's sphere. We consider among others cases when the dimension of the variety is higher than the expected one due to the presence of $(p + 1)$-tuples with non-trivial centralizers.

Let two Riemannian metrics $g$ and $\bar{g}$ on the torus $T^n$ have the same geodesics (considered as unparameterized curves). Then we can construct invariantly $n$ commuting differential operators of second order. The Laplacian $\Delta_g$ of the metric $g$ is one of these operators. For any $x \in T^n$, consider the linear transformation $G$ of $T_xT^n$ given by the tensor $g^{i\alpha}{\bar{g}}_{{\alpha}j}$. If all eigenvalues of $G$ are different at one point of the torus then they are different at every point; the operators are linearly independent and we can globally separate the variables in the equation $\Delta_gf = {\mu}f$ on this torus.

The classical problem of a free rigid body motion is considered within Kostant–Souriau prequantization programme supplemented by the semi-classical Bohr–Sommerfeld quantization rules. The results are two implicit formulae for the energy spectrum which are valid in some intervals defined by the total angular momentum.

Low-dimensional topology has experienced a number of revolutionary upheavals in the past twenty years. For many of these the seeds of the revolution were sown in theoretical physics and, more particularly, in the work of Edward Witten. The most recent such event occurred in 1994 when Witten suggested that the topological information about smooth 4-manifolds contained in the Donaldson invariants should also be contained in the much simpler and now famous Seiberg–Witten invariants. This lecture will provide an informal survey of some of the background behind the conjecture and how it came to be made.

This article is a survey which presents the essential ideas of “Observer Theory”, a formal theory of perception, developed since the late 80’s by Bruce Bennett and Donald Hoffman (both at U.C. Irvine) and myself. First I present the structure of an observer and one type of a framework, within which interactions between observers may be studied. Then I discuss the kinds of dynamics that can arise from such a framework, and how the dynamics can give rise to higher-level or “specialized” observers. Finally I indicate briefly what this says about “true” perception (i. e., perception adapted to the “world” the observer framework is in) and some possible ramifications which could lead to a deeper understanding of the origin of quantum systems and measurement theory. The general reference for this work is Bennett, Hoffman and Prakash [1].

The Maxwell–Bloch equations with one quadratic control about $Ox_1$ axis are introduced and some of their dynamical and geometrical properties are pointed out.

The local Lie structure of the orientation-reversing involutions on $\mathbb{R}^3$ is used to construct a family of orthogonally invariant operators that produce all formal solutions, up to biharmonic equivalence, of Navier’s equation for elastic equilibrium. In this construction the value of Poisson’s ratio associated with each solution is determined by the hyperbolic geometry of $sl_2(\mathbb{R})$. Empirically feasible values of the ratio are associated with ‘spacelike’ operators whereas values outside of this range are associated with ‘timelike’ operators.

The problem of diagonalization of Hamiltonians of $N-$dimensional boson systems by means of time-dependent canonical transformations (CT) is considered, the case of quadratic Hamiltonians being treated in greater detail. The unitary generator of time-dependent CT which can transform any Hamiltonian to that of a system of uncoupled stationary oscillators is constructed. The close relationship between methods of canonical transformations, time-dependent integrals of motion and dynamical symmetry is noted.

The diagonalization and symplectic properties of the uncertainty matrix for $2N$ canonical observables are studied. It is shown that the normalized uncertainty matrix is symplectic for the squeezed multimode Glauber coherent states and for the squeezed Fock states with equal photon numbers in each mode. The Robertson uncertainty relation for the dispersion matrix of canonical observables is shown to be minimized in squeezed coherent states only.