Moving Coframes: II. Regularization and Theoretical Foundations

The primary goal of this paper is to provide a rigorous theoretical justification of Cartans method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.     

Curves in Affine and Semi-Euclidean Spaces

In this paper we define centroaffine invariant arc length and curvature functions of a curve in affine n-space. Then we consider the properties and relations of the curves in affine space and Semi-Euclidean space. Using these notions and conclusions, by solving certain differential equations, we give some examples and classifications of the curves in affine 2-space and 3-space.     

Vassiliev invariants and finite-dimensional approximations of the euler equation in magnetohydrodynamics

We consider Hamiltonian systems that correspond to Vassiliev invariants defined by Chen’s iterated integrals of logarithmic differential forms. We show that Hamiltonian systems generated by first-order Vassiliev invariants are related to the classical problem of motion of vortices on the plane. Using second-order Vassiliev invariants, we construct perturbations of Hamiltonian systems for the classical problem of n vortices on the plane. We study some dynamical properties of these systems.     

A hierarchy of submodels of differential equations

We consider a system of differential equations admitting a group of transformations. The Lie algebra of the group generates a hierarchy of submodels. This hierarchy can be chosen so that the solutions to each of submodels are solutions to some other submodel in the same hierarchy. For this we must calculate an optimal system of subalgebras and construct a graph of embedded subalgebras and then calculate the differential invariants and invariant differentiation operators for each subalgebra. The invariants of a superalgebra are functions of the invariants of the algebra. The invariant differentiation operators of a superalgebra are linear combinations of invariant differentiation operators of a subalgebra over the field of invariants of the subalgebra. The comparison of the representations of group solutions gives a relation between the solutions to the models of the superalgebra and the subalgebra. Some examples are given of embedded submodels for the equations of gas dynamics.     

Differential invariants for systems of linear hyperbolic equations

In this paper we consider a general class of systems of two linear hyperbolic equations. Motivated by the existence of the Laplace invariants for the single linear hyperbolic equation, we adopt the problem of finding differential invariants for the system. We derive the equivalence group of transformations for this class of systems. The infinitesimal method, which makes use of the equivalence group, is employed for determining the desired differential invariants. We show that there exist four differential invariants and five semi-invariants of first order. Applications of systems that can be transformed by local mappings to simple forms are provided.     

Maurer–Cartan forms and the structure of Lie pseudo-groups

This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudo-group on submanifolds. The third paper [61] will apply Gr?bner basis methods to prove a fundamental theorem on the freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.     

Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

On Affine Surface That can be Completed by A Smooth Curve

In C⁶, we consider a non linear system of differential equations with four invariants: two quadrics, a cubic and a quartic. Using Enriques-Kodaira classification of algebraic surfaces, we show that the affine surface obtained by setting these invariants equal to constants is the affine part of an abelian surface. This affine surface is completed by gluing to it a one genus 9 curve consisting of two isomorphic genus 3 curves intersecting transversely in 4 points.     

On linearization of hyperbolic equations using differential invariants

In this paper we consider the general class of hyperbolic equations uxt=F(x,t,u,ux,ut). We use equivalence transformations to derive differential invariants for this class and for two subclasses. Then we employ these invariants to construct equations that can be linearized via local mappings. Further applications of the differential invariants are given.     

Relative morsification theory

In this paper we develop a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known morsification results for non-isolated singularities and generalise them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalisations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalised jet-space we show how to interpret the newly defined invariants as certain intersection multiplicities; finally, we characterise which invariants can be interpreted as intersection multiplicities in the above mentioned generalised jet space.     

Some problems of the theory of plasticity for metals with strength differential

Within the framework of the problem of describing the plastic deformation of polycrystalline metals with strength differential, we consider problems of selection of universal constants and functions of the material that control the influence of the first and third invariants of the stress tensor. The possibility of using different types of experiments for the determination of these constants and verification of the theory is analyzed. It is shown that the traditionally used combined tension–torsion experiments on a thin-walled tube do not enable one to distinguish the influence of hydrostatic pressure and the Lode angle. The expedience of using biaxial tension and other experiments on simple and complex loading of a thin-walled tube by an axial force and internal pressure is justified.     

Projective equivalence of smooth functions on the plane

In this paper, we find differential invariants for the action of the projective group on the space of smooth functions on the plane and propose a classification of orbits of this action.     

Contact-equivalence problem for linear hyperbolic equations

We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. é. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 119–142, 2005.     

Invariants and Bonnet-type theorem for surfaces in ℝ4

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.     

Conics in the hyperbolic plane intrinsic to the collineation group

In the manner of Steiner??s interpretation of conics in the projective plane we consider a conic in a planar incidence geometry to be a pair consisting of a point and a collineation that does not fix that point. We say these loci are intrinsic to the collineation group because their construction does not depend on an imbedding into a larger space. Using an inversive model we classify the intrinsic conics in the hyperbolic plane in terms of invariants of the collineations that afford them and provide metric characterizations for each congruence class. By contrast, classifications that catalogue all projective conics intersecting a specified hyperbolic domain necessarily include curves which cannot be afforded by a hyperbolic collineation in the above sense. The metric properties we derive will distinguish the intrinsic classes in relation to these larger projective categories. Our classification emphasizes a natural duality among congruence classes induced by an involution based on complementary angles of parallelism relative to the focal axis of each conic, which we refer to as split inversion (Definition 5.3).     

Discrete Moving Frames on Lattice Varieties and Lattice-Based Multispaces

In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver’s one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.     

Differential invariants of a class of Lagrangian systems with two degrees of freedom

We consider systems of Euler–Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie?s infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.     

Geometric expansion,Lyapunov exponents and foliations

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an application, we show examples of systems with persistent non-absolute continuous center and weak unstable foliations. This generalizes the remarkable results of Shub and Wilkinson to cases where the center manifolds are not compact.     

Moving frames and singularities of prolonged group actions

The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie's theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of "totally singular points," and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vector-valued functions, and methods for the solution to Lie determinant equations.     

An analogue of the Dedekind–Rademacher sum and certain ray class invariants of real quadratic fields

In this paper, we consider a certain product of double sine functions as an analogue of the Dedekind–Rademacher sum. Its reciprocity formulas are established by decomposition of a certain double zeta function. As their applications, we reconstruct and refine a part of Arakawa?s work on ray class invariants of real quadratic fields, and prove directly explicit relations between various invariants which are defined in terms of the double sine function and are related to the Stark–Shintani conjecture. Moreover, in some examples, new expressions of the invariants are revealed. As two appendices, we give a new proof of Carlitz?s three-term relation for the Dedekind–Rademacher sum and a simple proof of Arakawa?s transformation formula for an analogue of the generalized Eisenstein series originated with Lewittes.