Nonholonomic connections with extended Galilean groups of transformations

We continue the investigation of regular representations of the extended Galilean group {ie463-01} acting on the smooth canonical fiber bundle π: R ⁿ⁺¹ → I R ⁿ . First-and second-order nonholonomic affine connections Γ ₁, Γ ₂, Γ _1,2 are constructed using the results presented in papers [3, 4, 5, 6].     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. III

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups (1, n) and construct (1, n)-invariant generalized differential-geometric connections. In both regular and special cases of the representations of the considered groups (1, n), we find all affine nonholonomic , and _1,2-connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations (1, n) and also describe the corresponding (1, n)invariant planar connections.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations

In the canonical smooth fiber bundles : ^{n+1 n} endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups and construct -invariant generalizations of differentiable connections. In both regular and special cases of the representations of the relevant groups , we found all the affine nonholonomic ₁-, ₂-, and _{1, 2}-connections of the first order (see [4]) possessing the local Lie groups of transformations and also described the respective -invariant planar connections.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. II

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups and construct -invariant generalizations of differentiable connections. In both regular and special cases of the representations of the relevant groups , we found all the affine nonholonomic -, -, and -connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations and also described the respective -invariant planar connections.     

On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections

We study the properties of real realizations of holomorphic linear connections over associative commutative algebras \(\mathbb{A}\) _m with unity. The following statements are proved.If a holomorphic linear connection ? on M _n over \(\mathbb{A}\) _m (m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ? of the Lie algebra of all affine vector fields of the space (M _mn ^? , ?^?) is no greater than (mn)² ? 2mn + 5, where m = dim_? \(\mathbb{A}\), \(n = dim_\mathbb{A} \) M _n , and ?^? is the real realization of the connection ?.Let ?^? =¹ ? ײ ? be the real realization of a holomorphic linear connection ? over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor ¹ R ≠ 0, ² R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space (M _2n ^? , ?^?) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( ^a M _n , ^a ?) (a = 1, 2).     

变质量非完整系统的形式不变性与Lie对称性

研究变质量非完整系统的形式不变性和Lie对称性．给出变质量非完整系统在无限小变换下形式不变性和Lie对称性的定义、判据及存在守恒量的定理,得到形式不变性和Lie对称性的关系,并举例说明结果的应用．     

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

The isometry groups of Riemannian orbifolds

We prove that the isometry group ?(\(\mathcal{N}\)) of an arbitrary Riemannian orbifold \(\mathcal{N}\), endowed with the compact-open topology, is a Lie group acting smoothly and properly on \(\mathcal{N}\). Moreover, ?(\(\mathcal{N}\)) admits a unique smooth structure that makes it into a Lie group. We show in particular that the isometry group of each compact Riemannian orbifold with a negative definite Ricci tensor is finite, thus generalizing the well-known Bochner’s theorem for Riemannian manifolds.     

Semigroups in symmetric Lie groups

Let G be a Lie group and L C G a Lie subgroup. We give necessary and sufficient conditions for a family of cosets of L to generate a subsemigroup with nonempty interior in G. We apply these conditions to symmetric pairs (G, L) where L is a subgroup of G such that Go C L C Gi and r is an involutive automorphism of G. As a consequence we prove that for several r the fixed point group GI is a maximal semigroup.     

Gauss-Manin connections for arrangements, III Formal connections

We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.

     

Constructing arithmetic subgroups of unipotent groups

Let G be a unipotent algebraic subgroup of some defined over . We describe an algorithm for finding a finite set of generators of the subgroup . This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.     

Symmetry of the Barochronous Motions of a Gas

We study the system of nonlinear differential equations which expresses the constancy of the algebraic invariants of the Jacobian matrix for smooth vector fields in three-dimensional space. This system is equivalent to the equations of gas dynamics which describe the barochronous motions of a gas (the pressure and density depend only on the time). We present the results of computation of the admissible local Lie group and construction of the general solution of the system. We mention a few new problems that arise here.     

Locally homogeneous affine connections on compact surfaces

Global properties of locally homogeneous and curvature homogeneous affine connections are studied. It is proved that the only locally homogeneous connections on surfaces of genus different from 1 are metric connections of constant curvature. There exist nonmetrizable nonlocally symmetric locally homogeneous affine connections on the torus of genus 1. It is proved that there is no global affine immersion of the torus endowed with a nonflat locally homogeneous connection into .

     

Lie algebras and separable morphisms in pro-affine algebraic groups

Let be an algebraically closed field of arbitrary characteristic, and let be a surjective morphism of connected pro-affine algebraic groups over . We show that if is bijective and separable, then is an isomorphism of pro-affine algebraic groups. Moreover, is separable if and only if (its differential) is surjective. Furthermore, if is separable, then .

     

Isoperimetry of nilpotent groups

This survey gives an overview of the isoperimetric properties of nilpotent groups and Lie groups. It discusses results for Dehn functions and filling functions as well as the techniques used to retrieve them. The content reaches from long standing results up to the most recent development.     

The independence of characters on non-abelian groups

We show that there are characters of compact, connected, non-abelian groups that approximate random choices of signs. The work was motivated by Kronecker's theorem on the independence of exponential functions and has applications to thin sets.

     

On e-cuspidal pairs of finite groups of exceptional Lie type

Let G be a simple, simply connected algebraic group of exceptional type defined over ${\mathbb{F}}_q$ with Frobenius endomorphism $F:G\to G$. Let $??q$ be a good prime for G. We determine the number of irreducible Brauer characters in the quasi-isolated ?-blocks of $G^F$. This is done by proving that generalized e-Harish-Chandra theory holds for the Lusztig series associated to quasi-isolated elements of $G^{?F}$.     

Extentions between admissible representations of isometry groups of regular trees

This is the continuation of our previous work (Demir, Geom. Dedicata 105, 189–207, 2004). In this paper, we study extensions between admissible representations of the isometry groups of regular trees. We use the results of Demir, (Geom. Dedicata 105, 189–207, 2004) to show, following Schneider and Stuhler (J. Reine. Angew. Math. 436, 19–32, 1993) in the p-adic case, that extensions between admissible representations of G are finite-dimensional.