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The generalized Finsler geometry, as well as Finsler geometry, is a generalization of Riemann geometry. The generalized Finsler geometry can be endowed with the Cartan connection. The generalized Finsler geometry and its Cartan connection…
We define a notion of equivariant non-degeneracy of $G$-maps to introduce the class of equivariantly non-degenerate flows on smooth compact manifolds with compact Lie group action. We prove genericity of this class and use this result to…
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in…
The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively investigated in the past decades. A particular topic of interest has been the study of the asymptotics of their projections. Apart from their intrinsic interest,…
The main result of this paper is a convexity theorem for momentum mappings of certain hamiltonian actions of noncompact semisimple Lie groups. The image is required to fall within a certain open subset D of the (dual of the) Lie algebra,…
It is surprising, but an established fact that the field of Elementary Geometry referring to normed spaces (= Minkowski spaces) is not a systematically developed discipline. There are many natural notions and problems of elementary and…
Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the…
In this paper we review the known facts on isometries of Minkowski geometries and prove some new results on them. We give the normal forms of two special classes of operators and also characterize the isometry group of Minkowski $3$-spaces…
The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski…
Given an increasing odd homeomorphism $\sigma$ : R $\rightarrow$ R, the two bijective maps h $\sigma$ , v $\sigma$ : R 2 $\rightarrow$ R 2 dened by h $\sigma$ (x, y) = (x + $\sigma$ --1 (y), y) and v $\sigma$ (x, y) = (x, $\sigma$(x) + y).…
In this paper we introduce the notion of orbit equivalence for semigroup actions and the concept of generalized linear control system on smooth manifold. The main goal is to prove that, under certain conditions, the semigroup system of a…
We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\bf R})\ltimes {\bf R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\bf R})\ltimes {\bf R}^3$,…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to…
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a…
As a sequel to (Berman, 2008a), we show that the rotation of the Universe can be dealt by generalised Gaussian metrics, defined in this paper. Robertson-Walker's metric has been employed with proper-time, in its standard applications; the…
We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In…
The classical isoperimetric inequality can be extended to a general normed plane. In the Euclidean plane, the defect in the isoperimetric inequality can be calculated in terms of the signed areas of some singular sets. In this paper we…
The second Poincar\'e kinematical group serves as one of new ones in addition to the known possible kinematics. The geometries with the second Poincar\'e symmetry is presented and their properties are analyzed. On the geometries, the new…