Differential-geometric structures on manifolds

The work is an analytically systematic exposition of modern problems in the investigation of differentiable manifolds and the geometry of fields of geometric objects on such manifolds.Translated from Itogi Nauki i Tekhniki, Problemy Geometrii, Vol. 9, pp. 5–246.The authors thank the members of the Geometry Seminar Committee: Professor B. L. Laptev, Professor A. P. Norden, Professor V. T. Bazylev, and Professor V. V. Ryzhkov, who took part in discussing the topics and structure of the book, for their suggestions and remarks. The authors also express their gratitude to the scientific co-worker of the Department of Mathematics of VINITI, Z. A. Izmailova, who provided much assistance in the preparation of the book.     

Differential-geometric structures in operads

We describe certain structures of formal differential geometry in terms of the theory of operads and introduce group structures, Lie-algebra structures, exponential mappings, and an analog of the de Rham complex.     

Differential-geometric structures defining higher order contact transformations

The paper is devoted to the study of differential-geometric structures generated by Lie-Bäcklund transformations (or, what is the same, higher order contact transformations), which are a special case of diffeomorphisms between two manifolds of holonomic jets of sections. We study the structure of the fundamental object of a second order contact diffeomorphism (2-diffeomorphism). We also consider the case when a 2-diffeomorphism is given by explicit equations connecting local coordinates of 2-jet manifolds and establish conditions under which 2-diffeomorphisms defined by explicit equations are contact diffeomorphisms.     

Differential-geometric structures on generalized Reidemeister and Bol three-webs

In this paper, we present the main results of the study of multidimensional three-websW(p, q, r) obtained by the method of external forms and moving Cartan frame. The method was developed by the Russian mathematicians S. P. Finikov, G. F. Laptev, and A. M. Vasiliev, while fundamentals of differential-geometric (p, q, r)-webs theory were described by M. A. Akivis and V. V. Goldberg. Investigation of (p, q, r)-webs, including algebraic and geometric theory aspects, has been continued in our papers, in particular, we found the structure equations of a three-web W(p, q, r), where p = λl, q = λm, and r = λ(l + m − 1). For such webs, we define the notion of a generalized Reidemeister configuration and proved that a three-web W(λl, λm, λ(l + m − 1)), on which all sufficiently small generalized Reidemeister configurations are closed, is generated by a λ-dimensional Lie group G. The structure equations of the web are connected with the Maurer–Cartan equations of the group G. We define generalized Reidemeister and Bol configurations for three-webs W(p, q, q). It is proved that a web W(p, q, q) on which generalized Reidemeister or Bol configurations are closed is generated, respectively, by the action of a local smooth q-parametric Lie group or a Bol quasigroup on a smooth p-dimensional manifold. For such webs, the structure equations are found and their differential-geometric properties are studied.     

Gauge theories as matrix models

We discuss the relation between the Seiberg-Witten prepotentials, Nekrasov functions, and matrix models. On the semiclassical level, we show that the matrix models of Eguchi-Yang type are described by instantonic contributions to the deformed partition functions of supersymmetric gauge theories. We study the constructed explicit exact solution of the four-dimensional conformal theory in detail and also discuss some aspects of its relation to the recently proposed logarithmic beta-ensembles. We also consider “quantizing” this picture in terms of two-dimensional conformal theory with extended symmetry and stress its difference from the well-known picture of the perturbative expansion in matrix models. Instead, the representation of Nekrasov functions using conformal blocks or Whittaker vectors provides a nontrivial relation to Teichmüller spaces and quantum integrable systems.     

Gauge transformations in constrained theories

A method is proposed for constructing infinitesimal gauge transformations for an arbitrary degenerate Lagrangian without restrictions on the constraint algebra. It is thereby shown that in the general case degeneracy of a theory is due to its gauge invariance.Joint Institute for Nuclear Research; State University, Tbilisi. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 1, pp. 66–73, January, 1995.     

Differential-geometric structures in the theory of two-dimensional integrable equations

The gauge representations of the integrable generalization of the Heisenberg magnet in (2+1)-dimensional space-time are interpreted in terms of topological charge. Restrictions on the class of solutions to the equation for a two-dimensional magnet are described for which it becomes gauge-equivalent to the Davy-Stuartson equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 75–81, 1991.The author is grateful to P. P. Kulish for stimulating discussions.     

Universal regularizations. II. Gauge theories

V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 2, pp. 195–204, August, 1990.     

Gauge theories for gravity on a line

In Memoriam M. C. Polivanov     

Differential-geometric Poisson brackets on a lattice

M. V. Lomonosov Moscow State Univeristy. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 2, pp. 57–59, April–June, 1989.     

Direct product decompositions of structures and theories

In this article we present new methods in the theory of direct product decompositions of arbitrary structures.These methods are based on works of the author on the logic of product structures and on certain product-theoretic operations on sentences. In addition, some results included here rely on certain combinatorial properties and in such questions we apply an extension of König Infinity Lemma, also due to the author.Presented by R. McKenzie.     

Unstable structures definable in o-minimal theories

Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in N^eq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.     

Differential-geometric and group-theoretical methods in optimal control theory

Problems arising in the geometry of attainable sets of nonlinear control systems are considered. Old and new results are discussed.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 14, pp. 3–56, 1983.     

Definable one dimensional structures in o-minimal theories

Let N be a structure definable in an o-minimal structure M and p ∈ S _N (N), a complete N-1-type. If dim _M (p) = 1, then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N-definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to a generic type of an N-definable real closed field.