Classification of Material <Emphasis Type="Italic">G</Emphasis>-Structures

A geometrical interpretation of the G-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained.     

On the torsion of linear higher order connections

For a linear r-th order connection on the tangent bundle we characterize geometrically its integrability in the sense of the theory of higher order G-structures. Our main tool is a bijection between these connections and the principal connections on the r-th order frame bundle and the comparison of the torsions under both approaches.     

Quaternionic and para-quaternionic <Emphasis Type="Italic">CR</Emphasis> structure on (4n+3)-dimensional manifolds

We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω₁,ω₂,ω₃) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.     

Compatible Structures on Lie Algebroids and Monge-Ampère Operators

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between PN-, P Ω- and Ω N-structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampère operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold.     

Some groups of type VF

A group is of type VF if it has a finite-index subgroup which has a finite classifying space. We construct groups of type VF in which the centralizers of some elements of finite order are not of type VF and groups of type VF containing infinitely many conjugacy classes of finite subgroups. It follows that a group G of type VF need not admit a finite-type universal proper G-space. We construct groups G for which the minimal dimension of a universal proper G-space is strictly greater than the virtual cohomological dimension of G. Each of our groups embeds in GL _m (ℤ) for sufficiently large m. Some applications to K-theory are also considered. Oblatum 26-IV-2001 & 3-VII-2002?Published online: 10 October 2002     

Finite-type invariants for graphs and graph reconstructions

Many of the fundamental open problems in graph theory have the following general form: How much information does one need to know about a graph G in order to determine G uniquely. In this article we investigate a new approach to this sort of problem motivated by the notion of a finite-type invariant, recently introduced in the study of knots. We introduce the concepts of vertex-finite-type invariants of graphs, and edge-finite-type invariants of graphs, and show that these sets of functions have surprising algebraic properties. The study of these invariants is intimately related with the classical vertex- and edge-reconstruction conjectures, and we demonstrate that the algebraic properties of the finite-type invariants lead immediately to some of the fundamental results in graph reconstruction theory.     

Coordinate finite-type submanifolds

A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of . We prove that the compact coordinate finite-type submanifolds are minimal submanifolds of quadratic hypersurfaces of Euclidean spaces. Moreover, we classify the compact coordinate finite-type submanifolds of codimension 2.     

Cartan–Laptev method in the theory of multidimensional three-webs

We show how the Cartan–Laptev method that generalizes Elie Cartan’s method of external forms and moving frames is applied to the study of closed G-structures defined by multidimensional three-webs formed on a C ^s -smooth manifold of dimension 2r, r ≥ 1, s ≥ 3, by a triple of foliations of codimension r. We say that a tensor T belonging to a differential-geometric object of order s of a three-web W is closed if it can be expressed in terms of components of objects of lower order s. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed G-structure. It follows from our results that the G-structure associated with a hexagonal three-web W is a closed G-structure of class 4. It is proved that basic tensors of a three-web W belonging to a differential-geometric object of order s of the web can be expressed in terms of an s-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web W with closed G-structure of class s is completely defined by an s-jet of this expansion. We also consider webs with one-digit identities of kth order in their coordinate loops and find the conditions for these webs to have the closed G-structure.     

A differential-geometric criterion of the kinematic integrability of nonlinear differential equations

A theorem on the existence of a G-representation and a differential-geometric criterion of the kinematic integrability for nonlinear differential equations from the Λ²-G-classes is proved. Examples of zero-curvature representations and metrics for some equations of mathematical physics are presented. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

Half-flat structures on decomposable lie groups

Half-at SU(3)-structures are the natural initial values for Hitchin’s evolution equations whose solutions define parallel G₂-structures. Together with the results of [SH], the results of this article completely solve the existence problem of left-invariant half-at SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras, which refines and clarifies the existing classification of five-dimensional Lie algebras.     

The co-degrees of irreducible characters

LetG be a finite group. The co-degree of an irreducible character χ ofG is defined to be the number |G|/χ(1). The set of all prime divisors of all the co-degrees of the nonlinear irreducible characters ofG is denoted by Σ(G). First we show that Σ(G)=π(G) (the set of all prime divisors of |G|) unlessG is nilpotent-by-abelian. Then we make Σ(G) a graph by adjoining two elements of Σ(G) if and only if their product divides a co-degree of some nonlinear character ofG. We show that the graph Σ(G) is connected and has diameter at most 2. Additional information on the graph is given. These results are analogs to theorems obtained for the graph corresponding to the character degrees (by Manz, Staszewski, Willems and Wolf) and for the graph corresponding to the class sizes (by Bertram, Herzog and Mann). Finally, we investigate groups with some restriction on the co-degrees. Among other results we show that ifG has a co-degree which is ap-power for some primep, then the corresponding character is monomial andO _p (G)≠1. Also we describe groups in which each co-degree of a nonlinear character is divisible by at most two primes. These results generalize results of Chillag and Herzog. Other results are proved as well. The paper was written during this author’s visit at the Technion and the University of Tel Aviv. He would like to thank the departments of mathematics at the Technion and the University of Tel Aviv for their hospitality and support.     

General Natural $$(\alpha ,\varepsilon )$$-Structures

We study in a unified way the \((\alpha ,\varepsilon )\)-structures of general natural lift type on the tangent bundle of a Riemannian manifold. We characterize the general natural \(\alpha \)-structures on the total space of the tangent bundle of a Riemannian manifold, and provide their integrability conditions (the base manifold is a space form and some involved coefficients are rational functions of the other ones). Then, we characterize the two classes (with respect to the sign of \(\alpha \varepsilon \)) of \((\alpha ,\varepsilon )\)-structures of general natural type on TM. The class \(\alpha \varepsilon =-1\) is characterized by some proportionality relations between the coefficients of the metric and those of the \(\alpha \)-structure, and in this case, the structure is almost Kählerian if and only if the first proportionality factor is the derivative of the second one. Moreover, the total space of the tangent bundle is a Kähler manifold if and only if it depends on three coefficients only (two coefficients of the integrable \(\alpha \)-structure and a proportionality factor).     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.     

The Automorphism Groups of Finite Type G-Structures on Orbifolds

The automorphism group of a G-structure of finite type and order k on a smooth n-dimensional orbifold is proved to be a Lie group of dimension n+dim(g+g ₁+...+g _k-1), where g _i is the ith prolongation of the Lie algebra g of a given group G. This generalizes the corresponding result by Ehresmann for finite type G-structures on manifolds. The presence of orbifold points is shown to sharply decrease the dimension of the automorphism group of proper orbifolds. Estimates are established for the dimension of the isometry group and the dimension of the group of conformal transformations of Riemannian orbifolds, depending on the types of orbifold points.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

On pointwise bounded approximation

Let G be an open subset in the extended complex plane and let A（G） denote the algebra of all functions analytic on G and continuous on G. We call a domain multi-nicely connected if there is a circular domain W and a conformal map ~ from W onto G such that the boundary value function of φ is univalent almost everywhere with respect to the arclength on aW. Suppose that every component of G is finitely connected and none of the components of G have single point boundary components. We show that for every bounded analytic function on G to be the pointwise limit of a bounded sequence of functions in A（G）, it is necessary and sufficient that each component of G is multi-nicely connected and the harmonic measures of G are mutually singular. This generalizes the corresponding result of Davie for the case when the components of G are simply connected.     

<Emphasis Type="Italic">∂</Emphasis>-operations and <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">V</Emphasis></Subscript>-fields

The hyperoperations, called theta-operations （δ）, are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures.     

A rank-three condition for invariant (1, 2)-symplectic almost Hermitian structures on flag manifolds

This paper considers invariant (1, 2)-symplectic almost Hermitian structures on the maximal flag manifod associated to a complex semi-simple Lie group G. The concept of cone-free invariant almost complex structure is introduced. It involves the rank-three subgroups of G, and generalizes the cone-free property for tournaments related to 𝕊l (n,ℂ) case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant (1, 2)-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not B _l , l ≥ 3, G ₂ or F ₄. For B _l and F ₄ a close condition turns out to be sufficient. Received: 28 October 2001     

Finite solvable groups in which the Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups are either bicyclic or of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript>

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p ³ for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A₄-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p ² + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.     

On the stability of the index of unbounded nonlocal operators in Sobolev spaces

Unbounded operators corresponding to nonlocal elliptic problems on a bounded region G ⊂ ℝ² are considered. The domain of these operators consists of functions in the Sobolev space W ₂ ^m (G) that are generalized solutions of the corresponding elliptic equation of order 2m with the right-hand side in L ₂(G) and satisfy homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved that lower order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 116–135.