An uncertainty principle on the group of rigid motions of the plane

Let M(2) be the group of rigid motions of the plane. The Fourier transform and the Plancherel formula on M(2) can be explicitly given by the general group representation theory. Using this fact, we establish a kind of uncertainty principle on M(2). The result can easily be generalized to higher dimensional cases. An application of the result yields an uncertainty principle on the Euclidean spaces obtained by R. S. Strichartz.     

Polynomial alternatives for the group of affine motions

It is known that every polycyclic-by-finite group – even if it admits no affine structure – allows a polynomial structure of bounded degree. A major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of , in contrast to the group of affine motions, is no longer a finite dimensional Lie group. In this paper we construct a family of (finite dimensional) Lie groups, even linear algebraic groups, of polynomial diffeomorphisms, which we call weighted groups of polynomial diffeomorphisms. It turns out that every polycyclic-by-finite group admits a polynomial structure via these weighted groups; in the nilpotent (and other) case(s), we can sharpen, by specifying a nice set of weights, the existence results obtained in earlier work. We introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding Mal`cev completion. This, and other properties, provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e.g. in the work of Fried, Goldman and Hirsch. Received November 30, 1998; in final form March 10, 1999     

Dynamical invariants of rigid motions on the hyperbolic plane

This paper is devoted to the investigation of the geometry of left invariant metrics on the isometry group of the hyperbolic plane determined by the kinetic energy of a rigid particle system.Dedicated to the memory of Professor A. M. Vasil'ev (1923–1987)     

The Gauss-Bonnet defect of complex affine hypersurfaces

We study the loss of curvature at the “ends” of a hypersurface in the affine space and we express it in terms of singularities at infinity.     

Nonsimplicity of the group of unimodular automorphisms of the affine plane

It is proven that the group of automorphisms of the affine plane (as ao algebraic manifold), with Jacobian equal to 1, is not simple.Translated from Matematicheskie Zametki, Vol. 15, No. 2, pp. 289–293, February, 1974.     

Cauchy-Riemann equations in Minkowski plane

The properties of the symmetry and ordering of Minkowski plane are discussed by using hyperbolic imaginary unit and elliptic imaginary unit of Clifford algebra, and the representations of Cauchy-Riemann equations are given in Minkowski plane.     

On the affine deformation of a Minkowski norm

We prove that the affine deformation of a Minkowski norm is a Minkowski norm. Some important classical norms are derived by using the affine deformation recipe. Applications are done for some special Finsler manifolds.     

Equivalence theorems in affine differential geometry

In this paper we establish an affine equivalence theorem for affine submanifolds of the real affine space with arbitrary codimension. Next, this theorem is used to prove the classical congruence theorem for submanifolds of the Euclidean space, and to prove some results on affine hypersurfaces of the real affine space.Research Assistant of the National Fund for Scientific Research (Belgium).     

Some theorems in affine differential geometry

In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimensionn+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurfacex ₁ x ₂...x _n+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.The Project Supported by National Natural Science Foundation of China     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.     

On the properly discontinuous subgroups of affine motions

The fundamental groupΓ of a compact complete affine manifold is represented as an affine crystallographic subgroup of Aff(n). L.S.Auslander conjectured thatΓ is virtually solvable. Our purpose is to find the algebraic condition onΓ which leads affirmative answer to the conjecture.     

Minkowski type theorems for convex sets in cones

Acta Mathematica Hungarica - Minkowski’s classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface...     

Two Gauss-Bonnet and Poincaré-Hopf theorems for orbifolds with boundary

We generalize the Gauss-Bonnet and Poincaré-Hopf theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake, in which the local data (i.e. integral of the curvature in the case of the Gauss-Bonnet theorem and the index of the vector field in the case of the Poincaré-Hopf theorem) is related to Satake's orbifold Euler-Satake characteristic, a rational number which depends on the orbifold structure.For the second pair of generalizations, we use the Chen-Ruan orbifold cohomology to express the local data in a way which can be related to the Euler characteristic of the underlying space of the orbifold.     

Some equivalence theorems in affine hypersurface theory

Equivalence problems in affine geometry of hypersurfaces with type number greater than one are considered.Research supported by A. v. Humboldt-Stifung.     

Invariant subspaces of some functional spaces on the group of motions of the Euclidean plane

Petrozavodsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 3, pp. 135–146, May–June, 1990.