Local Diffeomorphisms Research Articles

On global invertibility of semi-algebraic local diffeomorphisms

On global invertibility of semi-algebraic local diffeomorphisms

A weight-homogenous condition to the real Jacobian conjecture in

AbstractIt is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

A new class of non-injective polynomial local diffeomorphisms on the plane

In this short note we provide the example with the lowest degree known so far of a non-injective local polynomial diffeomorphism F=(p,q):R2→R2. In our example p has degree 10 and q has degree 15, rather than 10 and 25, respectively, known up to now as the smallest degrees for the coordinates of F. Our construction was based on S. Pinchuk celebrated counterexample to the real Jacobian conjecture.

On a generalization of the Liouville conformal theorem to general rigid geometric structures

For any \(n\ge 3\), the classical Liouville theorem asserts that any local conformal diffeomorphism defined on a connected open subset of \({\mathbb {S}}^n\) can be uniquely extended to a M\(\ddot{\mathrm{o}}\)bius transformation. In this article, we formulate and prove a generalization of this theorem to general rigid geometric structures.

3+1 formulation of the standard model extension gravity sector

We present a 3+1 formulation of the effective field theory framework called the Standard-Model Extension in the gravitational sector. The explicit local Lorentz and diffeomorphism symmetry breaking assumption is adopted and we perform a Dirac-Hamiltonian analysis. We show that the structure of the dynamics presents significant differences from General Relativity and other modified gravity models. We explore Hamilton's equations for some special choices of the coefficients. Our main application is cosmology and we present the modified Friedmann equations for this case. The results show some intriguing modifications to standard cosmology. In addition, we compare our results to existing frameworks and models and we comment on the potential impact to other areas of gravitational theory and phenomenology.

Backgrounds in gravitational effective field theory

Effective field theories describing gravity coupled to matter are investigated, allowing for operators of arbitrary mass dimension. Terms violating local Lorentz and diffeomorphism invariance while preserving internal gauge symmetries are included. The theoretical framework for violations of local Lorentz and diffeomorphism invariance and associated conceptual issues are discussed, including transformations in curved and approximately flat spacetimes, the treatment of various types of backgrounds, the implications of symmetry breaking, and the no-go constraints for explicit violation in Riemann geometry. Techniques are presented for the construction of effective operators, and the possible terms in the gravity, gauge, fermion, and scalar sectors are classified and enumerated. Explicit expressions are obtained for terms containing operators of mass dimension six or less in the effective action for General Relativity coupled to the Standard Model of particle physics. Special cases considered include Einstein-Maxwell effective field theories and the limit with only scalar coupling constants.

LOSIK CLASSES FOR CODIMENSION-ONE FOLIATIONS

AbstractFollowing Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

On the liftability of expanding stationary measures

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs–Markov–Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding.

Closed Affine Manifolds with an Invariant Line

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.

Minimal Projectivity Condition for a Smooth Mapping and the Gronwall Problem

In this paper, the following assertion is proved: Let GW and $$ \tilde{GW} $$ be Grassmannian three-webs defined respectively in domains D and $$ \tilde{D} $$ of a Grassmannian manifold of straight lines of the projective space Pr+1; Φ : D → $$ \tilde{D} $$ be a local diffeomorphism that maps foliations of the web GW to foliations of the web $$ \tilde{GW} $$. Then Φ maps bundles of lines to bundles of lines, i.e., induces a point transformation, which is a projective transformation. In the case where r = 1, the proof is much more complicated than in the multidimensional case. In the case where r = 1, the dual theorem is formulated as follows: Let LW be a rectilinear three-web on a plane, i.e., three families of lines in the general position, and let this web be not regular, i.e., not locally diffeomorphic to a three-web formed by three families of parallel straight lines. Then each local diffeomorphism that maps a three-web LW to another rectilinear three-web $$ \tilde{LW} $$ is a projective transformation. As a consequence, we obtain the positive solution of the Gronwall problem (Gronwall, 1912): If W is a linearizable irregular three-web and θ and $$ \tilde{\theta} $$ are local diffeomorphisms that map the three-web W to some rectilinear three-webs, then $$ \tilde{\theta} $$ = π ° θ, where π is a projective transformation.

A Global Diffeomorphism Theorem for Fréchet Spaces

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fr\'{e}chet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fr\'{e}chet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fr\'{e}chet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.

Decoupled Intensity-Based Nonmetric Visual Servo Control

This brief addresses the problem of vision-based robot control where the equilibrium state is defined via a goal image. Specifically, we consider the class of intensity-based nonmetric solutions, which provide for high accuracy, versatility, and robustness. Existing techniques within that class present a fully coupled translational and rotational control error dynamics, what increases analysis complexity and may degrade system performance. This brief proposes a new intensity-based nonmetric visual servoing technique that decouples the translational control error dynamics, regardless of the observed object characteristics, camera displacements, and their relative poses. The obtained system is thus lower triangular in the general case. For some practical cases, the proposed general technique leads to the Grail of a fully decoupled (i.e., strictly diagonal) linear dynamics. The theoretical analysis of local diffeomorphism, local exponential stability, and those decoupling properties is provided. Improved performances are also experimentally confirmed using a 6-DoF robotic arm in various positioning and tracking tasks.

A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

Local asymptotic stabilizability is a topic of great theoretical interest and practical importance. Broadly, if a system \(\dot {x} = f(x,u)\) is locally asymptotically stabilizable, we are guaranteed a feedback controller u(x) that forces convergence to an equilibrium for trajectories initialized sufficiently close to it. A necessary condition was given by Brockett: such controllers exist only when f is open at the equilibrium. Recently, Gupta, Jafari, Kipka, and Mordukhovich considered a modification to this condition, replacing Brockett’s topological openness by the linear openness property of modern variational analysis. In this paper, we show that, under the linear openness assumption, there exists a local diffeomorphism rendering the system exponentially stabilizable by means of continuous stationary feedback laws. Introducing a transversality property and relating it to the above diffeomorphism, we prove that linear openness and transversality on a punctured neighborhood of an equilibrium is sufficient for local exponential stabilizability of systems with a rank-deficient linearization. The main result goes beyond the usual Kalman and Hautus criteria for the existence of exponential stabilizing feedback laws, since it allows us to handle systems for which exponential stabilization is achieved through higher order terms. However, it is implemented so far under a rather restrictive row-rank conditions. This suggests a twofold approach to the use of these properties: a pointwise version is enough to ensure stability via the linearization, while a local version is enough to overcome deficiencies in the linearization.

Homogeneous models for Levi degenerate CR manifolds

We extend the notion of a fundamental negatively Z-graded Lie algebra mx=⊕p≤−1mxp associated to any point of a Levi nondegenerate Cauchy-Riemann (CR) manifold to the class of k-nondegenerate CR manifolds (M,D,J) for all k≥2 and call this invariant the core at x∈M. It consists of a Z-graded vector space mx=⊕p≤k−2mxp of height k−2 endowed with the natural algebraic structure induced by the Tanaka and Freeman sequences of (M,D,J) and the Levi forms of higher order. In the case of CR manifolds of hypersurface type, we propose a definition of a homogeneous model of type m, that is, a homogeneous k-nondegenerate CR manifold M=G/Go with core m associated with an appropriate Z-graded Lie algebra Lie(G)=g=⊕gp and subalgebra Lie(Go)=go=⊕gop of the nonnegative part ⊕p≥0gp. It generalizes the classical notion of Tanaka of homogeneous models for Levi nondegenerate CR manifolds and the tube over the future light cone, the unique (up to local CR diffeomorphisms) maximally homogeneous 5-dimensional 2-nondegenerate CR manifold. We investigate the basic properties of cores and models and study the 7-dimensional CR manifolds of hypersurface type from this perspective. We first classify cores of 7-dimensional 2-nondegenerate CR manifolds up to isomorphism and then construct homogeneous models for seven of these classes. We finally show that there exists a unique core and homogeneous model in the 3-nondegenerate class.

Covariant Hamiltonian for gravity coupled top-forms

We review the covariant canonical formalism initiated by D'Adda, Nelson and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $\phi$. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" $d\phi$ and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in this framework, and a generalized Noether theorem is derived, both for global and local symmetries. We apply the formalism to vielbein gravity in $d=3$ and $d=4$. In the $d=3$ theory we can define form-Dirac brackets, and use an algorithmic procedure to construct the canonical generators for local Lorentz rotations and diffeomorphisms. In $d=4$ the canonical analysis is carried out using FPB, since the definition of form-Dirac brackets is problematic. Lorentz generators are constructed, while diffeomorphisms are generated by the Lie derivative. A "doubly covariant" hamiltonian formalism is presented, allowing to maintain manifest Lorentz covariance at every stage of the Legendre transformation. The idea is to take curvatures as "velocities" in the definition of momenta.

Kinematical gravitational charge algebra

When formulated in terms of connection and coframes, and in the time gauge, the phase space of general relativity consists of a pair of conjugate fields: the flux 2-form and the Ashtekar connection. On this phase-space, one has to impose the Gauss constraints, the vector, and scalar Hamiltonian constraints. These are respectively generating local SU(2) gauge transformations, spatial diffeomorphisms, and time diffeomorphisms. We write the Gauss and space diffeomorphism constraints as conservation laws for a set of boundary charges, representing spin and momenta, respectively. We prove that these kinematical charges generate a local Poincar\'e ISU(2) symmetry algebra. This gives strong support to the recent proposal of Poincar\'e charge networks as a new realm for discretized general relativity [Classical Quantum Gravity 36, 195014 (2019)].

Lie groupoids of mappings taking values in a Lie groupoid

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current Lie groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper \'etale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current Lie groupoid as a current Lie algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators given by postcomposition with a fixed function, between manifolds of $C^\ell$-functions. Under natural hypotheses, these operators turn out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map. These results are new in their generality and of independent interest.

Two Variations on the Periscope Theorem

A spherical periscope in multi-dimensional space is a system of two ideal mirrors that reflect the rays emanating from a fixed point to the rays coming back to the same point, and a reversed periscope is a system of two mirrors that reflect the rays having a fixed direction to the rays having the opposite direction. We describe the local diffeomorphisms of the wave fronts (spherical, in the former, and flat, in the latter cases), induced by these 2-mirror systems.

Ergodicity of non-autonomous discrete systems with non-uniform expansion

We study the ergodicity of non-autonomous discrete dynamical systems with non-uniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of $ C^{1+\alpha} $ local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure. Moreover, we will also prove that every exact non-uniform expandable finitely generated semigroup action of conformal $ C^{1+\alpha} $ local diffeomorphisms of a compact manifold is Lebesgue ergodic.

To Millionshchikov’s Problem on the Baire Class of Central Exponents of Diffeomorphisms

It is shown that central exponents of a local diffeomorpliism of a Riemannian manifold treated as functions on the direct product of the manifold and the space of its local diffeomorphisms with C1-compact-open topology belong to the fourth Baire class.