Thurston [16] proved that the group Diff$^{\infty}(M)$ of a smooth manifold $M$ is perfect, which implies the first homology group is trivial. If $M$ has a geometric structure, then the first homology of the group of automorphisms of $M$ preserving the geometric structure is not necessarily trivial. There are many results concerning this field. In this paper, we shall summarize the results of the first homology groups of automorphisms of manifolds with geometric structure.

In this paper we will report on some of our recent results about compact spacelike hypersurfaces with spherical boundary in the Lorentz–Minkowski space $\mathbb{L}^{n+1}$. In particular we will prove that the only compact spacelike hypersurfaces in $\mathbb{L}^{n+1}$ with constant mean curvature and spherical boundary are the hyperplanar balls and the hyperbolic caps. As for the case of the scalar curvature, we will prove that the only compact spacelike hypersurfaces in $\mathbb{L}^{n+1}$ with nonzero constant scalar curvature and spherical boundary are the hyperbolic caps. Our approach is based on the use of several integral formulas, among them there are a volume formula and a flux formula.

The differential geometry of a normal red blood cell is treated using the Cassinian oval for modelling its profile. In this connection an explicit parametrization via Jacobian elliptic functions of the usual polar coordinates is found. The first and the second fundamental forms, and correspondingly, the Gaussian, mean, and principal curvatures, are derived. The integrals determining the volume, area, cross-section area, and circumference of a red blood cell are evaluated analytically and expressed in a form relevant to the sphere geometry via some correction factors. The free elastic energy $U,$ associated with the outer bilayer membrane of the cell is integrated and its scale dependence is established. A relation between $U$ and the surface area correction factor is determined. Approximate formulae, using elementary functions, that should be directly applicable to experimental data are developed.

Plots of these dimensionless parts of volume, area, cross-section area, and circumference are obtained. The sphericity index, homogeneity index, and volume/area ratio associated with the red cell geometry are derived in approximate forms as well.

Main features of the induced gauge field in the confining potential approach to constrained system are summarized. The relation of this approach to the coset space approach is mentioned. Properties of the induced gauge field coming from W-Z-W term in $SU (3)$ Skyrme–Witten model is examined.

The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method (ISM) is interesting and still an open problem. We show how the second-order reductions of the $N$-wave interactions related to low-rank simple Lie algebras can be embedded in the Weyl group of $\mathfrak{g}$. Some of the reduced systems find applications to nonlinear optics

Our previous results on the $N$-soliton interaction in the adiabatic approximation are extended. It is shown that the complex Toda chain (CTC) model is a universal one in the sense that it describes the $N$-soliton train interactions for all NLEE from the NLS hierarchy. We derive the perturbed CTC system and show that the small perturbations affect only the center of mass motion and the global phase of the $N$-soliton train. The peculiarities of the interactions in the nonadiabatic cases are outlined.

Higher Dolbeaut cohomology groups of the flag manifolds are explicitly constructed using the technique of the intertwining operators. The integral representation for the higher order harmonic forms is obtained

We prove, modulo a conjecture due to A. A. Bolibrukh, that every monodromy group in which the operators of local monodromy in their Jordan normal forms have Jordan blocks of size only $\leq 2$ can be realized by a Fuchsian system of linear differential equations on Riemann's sphere without additional apparent singularities. This implies that the Gauss-Manin system of a polynomial of two variables can always be made Fuchsian if a suitable basis in the cohomologies is chosen.

In this paper we introduce and define the quaternion; we give a brief introduction to its properties and algebra, and we show (what appears to be) its primary application—the quaternion rotation operator. The quaternion rotation operator competes with the conventional matrix rotation operator in a variety of rotation sequences.

In this paper we consider the problem of quantizing the geodesic motion on noncompact surfaces of constant negative curvature. This problem can be regarded as a model of multichannel quantum scattering. Knowing that the geodesic motion on such surfaces is chaotic, we examine how the chaos of the underlying classical dynamics manifests itself in the corresponding quantum system. We calculate the scattering matrix, and introduce the associated time delays. With the help of Selberg's trace formula we establish a connection between the classical periodic orbits and the quantum resonances and energy eigenvalues. Illustrative examples for a class of $\sum_{g,2}$ surfaces are given

The twistor integral representations of solutions of the Laplacian on the complex space are well-known. The purpose of this article is to generalize the results above to that of the sub-Laplacian on the odd-dimensional complex space with the standard contact structure.

Here we combine group-theoretical and differential-geometric techniques for considerations of minimal surface deformations in the ordinary three-dimensional space. This approach allows a consideration of a novel family of transformations generated by complex rotations. The resulting generalized deformations are compared with the well-known Bonnet and Goursat transformations and illustrated via the Schwarz skew quadrilateral to provide a clarification of their geometrical origin.

In this paper a generalization of the concept of action is considered. This notion is based on a new algebraic structure called generalized groups. An action is deduced by imposing an Abelian condition on a generalized group. Generalized actions on normal generalized groups are also considered

Dirac magnetic monopoles, which may or may not exist in nature, seem to exist everywhere in mathematics. They are in one-to-one correspondence with the natural connections on principal $U(1)$-bundles over $S^2$ and, moreover, appear as solutions to the field equations of $SU(2)$ Yang–Mills–Higgs theory on $\mathfrak{R}^3$ as well as Seiberg–Witten theory and its non-Abelian generalization on Minkowski space-time. This talk will present an informal survey of the situation.

The Green’s function for 5-dimensional counterpart of the MIC-Kepler problem (Kepler potential plus $SU(2)$ Yang-Mills instanton plus Zwanziger-like $1/R^2$ centrifugal term) is constructed on the basis of Green’s function for 8-dimensional harmonic oscillator.

The dynamics of the rigid body with three quadratic controls is discussed and some of its geometrical and dynamical properties are pointed out.

We define the contact Schwarzian derivatives $s_{[ij,k]}(\phi)$ for a contact transformation $\phi : \mathbb{K}^3 \rightarrow \mathbb{K}^3$. Using the contact Schwarzian derivatives as coefficients, we give a system of linear differential equations such that the solutions give the contact transformation.

The problem of two classical particles with central interaction on simply connected spaces of a constant curvature is considered from the invariant point of view. The Hamiltonian reduction method is used for exluding a movement of the system as a whole.

A reformulation of Rawnsley’s Kählerian coherent states (in the framework of geometric quantization) is used in order to investigate the interplay between their local and global properties (projective embeddings) and the relationship with Klauder quantization (via path integrals and the introduction of a metric on the classical phase space). A Klauder type formula is established for the projection operator onto the quantum Hilbert space (the kernel of a Bochner Laplacian) in terms of a phase space path integral. As a further application, a Riemann surface diastatic identity is derived, yielding, via Green function theory, a short proof of the Abel-Jacobi theorem (and conversely), together with some coherent state induced theta function identities.

The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schrödinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard $SU(1,1)$ and $SU(2)$ coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schrödinger inequality for the Hermitian components of the $su_q(1,1)$ ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form.

The generalization of a test particle motion in a central field of the two immovable point-like centers to the case of a constant curvature space, in the space of Lobachevsky, is studied in the paper. The bifurcation set in the plane of integrals of motion was constructed and the classification of the domains of possible motion was carried out. The Lagrange’s problem on the pseudosphere: a mass point motion under the action of attracting center field and the analogue of a constant homogeneous field in a constant curvature space, is studied as well.