Local Equivalence Problem Research Articles

Nonlinearizable CR Automorphisms for Polynomial Models in $$\mathbb C^N$$

The Lie algebra of infinitesimal CR automorphisms is a fundamental local invariant of a CR manifold. Motivated by the Poincaré local equivalence problem, we analyze its positively graded components, containing nonlinearizable holomorphic vector fields. The results provide a complete description of invariant weighted homogeneous polynomial models in $$\mathbb C^N$$ , which admit symmetries of degree higher than two. For homogeneous polynomial models, symmetries with quadratic coefficients are also classified completely. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove that such automorphisms arise from one common source, by pulling back via a holomorphic mapping a suitable symmetry of a hyperquadric in some (typically high dimensional) complex space.

Equivalence Conditions and Invariants for the General Form of Burgers\u2019 Equations

Equivalence of differential equations is one of the most important concepts in the theory of differential equations. In this paper, the moving coframe method is applied to solve the local equivalence problem for the general form of Burgers’ equation, which has two independent variables under action of a pseudo-group of contact transformations. Using this method, we found the structure equations and invariants of these equations, as a result some conditions for equivalence of them will be given.

Infinitesimal symmetries of weakly pseudoconvex manifolds

We consider weakly pseudoconvex hypersurfaces with polynomial models in \({\mathbb {C}}^N\) and their symmetry algebras. In the most prominent case of special models, given by sums of squares of polynomials, we give their complete classification. In particular, we prove that such manifolds do not admit any nonlinear symmetries, depending only on complex tangential variables, nor do they admit real or nilpotent linear symmetries. This leads to a sharp 2-jet determination result for local automorphisms. We also give partial results in the general case and a more detailed description of the graded components in complex dimension three. The results also provide an important necessary step for solving the local equivalence problem on such manifolds.

Local equivalence problem in hidden Markov model

In the hidden Markov process, there is a possibility that two different transition matrices for hidden and observed variables yield the same stochastic behavior for the observed variables. Since such two transition matrices cannot be distinguished, we need to identify them and consider that they are equivalent, in practice. We address the equivalence problem of hidden Markov process in a local neighborhood by using the geometrical structure of hidden Markov process. For this aim, we introduce a mathematical concept to express Markov process, and formulate its exponential family by using generators. Then, the above equivalence problem is formulated as the equivalence problem of generators. Taking this equivalence problem into account, we derive several concrete parametrizations in several natural cases.

Addendum to: Explicit Absolute Parallelism for 2-Nondegenerate Real Hypersurfaces $$M^5~\subset ~{\mathbb {C}}^{3}$$ of Constant Levi Rank 1

We study the local equivalence problem under biholomorphisms for five-dimensional real hypersurfaces $$M^5$$ of $$\mathbb {C}^3$$ which are 2-nondegenerate and of constant Levi rank 1. We find exactly two invariants, J and W, which are expressed explicitly in terms of the graphing function F of M, the vanishing of which gives a necessary and sufficient condition for M to be locally biholomorphic to a model hypersurface, the tube over the light cone. If one of the two invariants J or W does not vanish on M, we show that the equivalence problem under biholomorphisms reduces to an equivalence problem between $$\{e \}$$-structures, that is, we construct an absolute parallelism on M.

On the integrability problem for systems of partial differential equations in one unknown function, II

We continue here our discussion of Part I, [18], by examining the local equivalence problem for partial differential equations and illustrating it with some examples, since almost any integration process or method is actually a local equivalence problem involving a suitable model. We terminate the discussion by inquiring on non-integrable Pfaffian systems and on their integral manifolds of maximal dimension.

Differential Geometric Aspects of Causal Structures

This article is concerned with causal structures, which are defined as a field of tangentially non-degenerate projective hypersurfaces in the projectivized tangent bundle of a manifold. The local equivalence problem of causal structures on manifolds of dimension at least four is solved using Cartan's method of equivalence, leading to an $\{e\}$-structure over some principal bundle. It is shown that these structures correspond to parabolic geometries of type $(D_n,P_{1,2})$ and $(B_{n-1},P_{1,2})$, when $n\geq 4$, and $(D_3,P_{1,2,3})$. The essential local invariants are determined and interpreted geometrically. Several special classes of causal structures are considered including those that are a lift of pseudo-conformal structures and those referred to as causal structures with vanishing Wsf curvature. A twistorial construction for causal structures with vanishing Wsf curvature is given.

Explicit Absolute Parallelism for 2-Nondegenerate Real Hypersurfaces $$M^5 \subset \mathbb {C}^3$$ of Constant Levi Rank 1

We study the local equivalence problem under biholomorphisms for five-dimensional real hypersurfaces $$M^5$$ of $$\mathbb {C}^3$$ which are 2-nondegenerate and of constant Levi rank 1. We find exactly two invariants, J and W, which are expressed explicitly in terms of the graphing function F of M, the vanishing of which gives a necessary and sufficient condition for M to be locally biholomorphic to a model hypersurface, the tube over the light cone. If one of the two invariants J or W does not vanish on M, we show that the equivalence problem under biholomorphisms reduces to an equivalence problem between $$\{e \}$$-structures, that is, we construct an absolute parallelism on M.

Normal Form for Second Order Differential Equations

Applying methods of CR-geometry, we give a solution to the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a complete normal form (which is smooth or analytic respectively) for this class of ordinary differential equations (ODEs). The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE y ″ = 0. For a generic ODE, we also provide a unique (up to a discrete group action) normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold (1988). As another application of the normal form, we obtain distinguished curves associated with a differential equation that we call chains due to their analogy with the chains defined by Chern and Moser (Acta Math. 7;133:219–271).

Flat (2,3,5)-Distributions and Chazy's Equations

In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or $(2,3,5)$-distributions determined by a single function of the form $F(q)$, the vanishing condition for the curvature invariant is given by a 6$^{\rm th}$ order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7$^{\rm th}$ order nonlinear ODE described in Dunajski and Sokolov. We show that the 6$^{\rm th}$ order ODE can be reduced to a 3$^{\rm rd}$ order nonlinear ODE that is a generalised Chazy equation. The 7$^{\rm th}$ order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat $(2,3,5)$-distributions not of the form $F(q)=q^m$. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split $G_2$ as their group of symmetries.

Local nested transverse feedback linearization

We study two local feedback equivalence problems for a nonlinear control-affine system with two nested, controlled-invariant, embedded submanifolds in its state space. The first, less restrictive, result gives necessary and sufficient conditions for the dynamics of the system restricted to the larger submanifold and transversal to the smaller submanifold to be linear and controllable. This normal form facilitates designing controllers that locally stabilize the smaller set relative to the larger set. The second, more restrictive, result additionally imposes that the transversal dynamics to the larger set be linear and controllable. This result can simplify designing controllers to locally stabilize the larger submanifold. This is illustrated by sufficient conditions under which these normal forms can be used to locally solve a nested set stabilization problem.

The fundamental theorem for hypersurfaces in Heisenberg groups

We study the local equivalence problems of hypersurfaces in Heisenberg groups via Cartan’s method of moving frames and Lie groups. For non-singular points, we find a complete set of invariants. We also study the integrability conditions for hypersurfaces and show some kind of embedding theorems for surfaces satisfying the integrability conditions.

Point equivalence of functions on the 1-jet space J 1ℝ

The action of the pseudogroup of point transformations on the set of smooth functions on the contact 1-jet space J1ℝ is considered. Differential invariants of this action are found, and the problem of local point equivalence of regular smooth functions is solved.

Point equivalence of second-order ODEs: Maximal invariant classification order

We show that the local equivalence problem of second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also demonstrate that, modulo Cartan duality and point transformations, the Painlevé-I equation can be characterized as the simplest second-order ordinary differential equation belonging to the class of equations requiring 10th order jets for their classification.

Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy

Nurowski showed that any generic 2-plane field D on a 5-manifold M determines a natural conformal structure cD on M; these conformal structures are exactly those (on oriented M) whose normal conformal holonomy is contained in the (split, real) simple Lie group G2. Graham and Willse showed that for real-analytic D the same holds for the holonomy of the real-analytic Fefferman–Graham ambient metric of cD, and that both holonomy groups are equal to G2 for almost all D. We investigate here independently interesting 2-plane fields for which the associated holonomy groups are a proper subgroup of G2.Cartan solved the local equivalence problem for 2-plane fields D and constructed the fundamental curvature tensor A for these objects. He furthermore claimed to describe locally all D whose infinitesimal symmetry algebra has rank at least 6 and gave a local quasi-normal form, depending on a single function of one variable, for those that furthermore satisfy a natural degeneracy condition on A, but Doubrov and Govorov recently rediscovered a counterexample to Cartanʼs claim. We show that for all D given by Cartanʼs alleged quasi-normal form, the conformal structures cD induced via Nurowskiʼs construction are almost Einstein, that we can write their ambient metrics explicitly, and that the holonomy groups associated to cD are always the 5-dimensional Heisenberg group, which here acts indecomposably but not irreducibly. (Not all of these properties hold, however, for Doubrov and Govorovʼs counterexample.) We also show that the similar results hold for the related class of 2-plane fields defined on suitable jet spaces by ordinary differential equations z′(x)=F(y″(x)) satisfying a simple genericity condition.

Holomorphic Equivalence and Nonlinear Symmetries of Ruled Hypersurfaces in $\mathbb{C}^{2}$

We solve the local equivalence problem for ruled hypersurfaces in \(\mathbb {C}^{2}\) at a point of infinite type by giving a normal form. We describe all ruled hypersurfaces which admit nonlinear symmetries and give their complete characterization according to the dimension of their automorphism group. As an application, we give a bound for the dimension of the stability group for a family of nonminimal hypersurfaces.

Solving Local Equivalence Problems with the Equivariant Moving Frame Method

Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the equivariant moving frame constructions extend to submanifold jets where the pseudo-group does not act freely at any order. Once this is done, we review the solution to the local equivalence problem of submanifolds within the equivariant moving frame framework. This offers an alternative approach to Cartan's equivalence method based on the theory of G-structures.

Geometry of the local equivalence of states

We present a description of locally equivalent states in terms of symplectic geometry. Using the moment map between local orbits in the space of states and coadjoint orbits of the local unitary group, we reduce the problem of local unitary equivalence to an easy part consisting of identifying the proper coadjoint orbit and a harder problem of the geometry of fibers of the moment map. We give a detailed analysis of the properties of orbits of ‘equally entangled states’. In particular, we show connections between certain symplectic properties of orbits such as their isotropy and coisotropy with effective criteria of local unitary equivalence.

Equivariant moving frame method and the local equivalence of u xx = r(x, u, v, u x , v x ) under fiber-preserving transformations

We use the equivariant moving frame method to study the local equivalence problem of scalar control equations of the form uxx = r(x, u, v, ux, vx) under the pseudo-group of fiber-preserving transformations X = ϕ(x), U = β(x, u), V = α(x, u, v). Three typical branches of the equivalence problem are considered: the degenerate case, which contains the control systems with largest fiber-preserving symmetry group, the branch containing the Hilbert–Cartan equation, and the generic case.

Local equivalence problem for Levi flat hypersurfaces

In this paper we consider germs of smooth Levi flat hypersurfaces, under the following notion of local equivalence: S 1 ∼ S 2 if their one-sided neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a simple invariant for this relation, which allows us to prove some characterizations of triviality (i.e. equivalence to the hyperplane). Then, we employ the same invariant to construct infinitely many non-trivial classes, including an infinite family of non-equivalent hypersurfaces which are almost everywhere analytic.