Complete Vector Fields Research Articles

A global lifting for finite-dimensional Lie algebras of complete vector fields

Motivated by the so-called Lifting and Approximation Theorem by Rothschild and Stein, we consider a set of vector fields [Formula: see text] on a manifold [Formula: see text], and we study the problem of obtaining a global lifting of the [Formula: see text]’s to a system of generators of the Lie algebra [Formula: see text] of a Lie group [Formula: see text]. By assuming that the Lie algebra [Formula: see text] generated by [Formula: see text] is finite-dimensional and all of the [Formula: see text]’s are complete vector fields, but without the assumption that they satisfy Hörmander’s rank condition, we reduce the lifting problem to a result of Palais on integrability. This proves that any [Formula: see text] in [Formula: see text] is [Formula: see text]-related to a left invariant vector field [Formula: see text], where [Formula: see text] is a smooth map resulting from a right action [Formula: see text] of [Formula: see text] on [Formula: see text]. Both the lifting map [Formula: see text] and the lifting vector fields [Formula: see text] are globally defined, and our result generalizes the global Lifting obtained by Folland in the special case of dilation-invariant vector fields. According to Palais’ Integrability, the map [Formula: see text] is obtained via the flow of suitable vector fields on [Formula: see text] and the image set of [Formula: see text], namely, [Formula: see text], is the Sussmann orbit of [Formula: see text] through [Formula: see text]. Examples are provided, showing that a germ of [Formula: see text], obtained through the integration of [Formula: see text] and depending only on the Baker–Campbell–Hausdorff formula, is often sufficient to get the global lifting without the need of abstract results.

Smooth actions of connected compact Lie groups with a free point are determined by two vector fields

Consider a smooth action G×M→M of a compact connected Lie group G on a connected manifold M. Assume the existence of a point of M whose isotropy group has a single element (a free point). Then we prove that there exist two complete vector field X,X1 such that their group of automorphisms equals G regarded as a group of diffeomorphisms of M (the existence of a free point implies that the action of G is effective). Moreover, some examples of effective actions with no free point where this result fails are exhibited.

Hydrodynamic Killing vector fields on surfaces

Killing vector fields, which have their origins in Riemannian geometry, have recently garnered attention for their significance in understanding fluid flows on curved surfaces. Owing to the significance of behavior of fluid flows around the boundary and at infinity, in the context of fluid dynamics, Killing vector fields of interest should satisfy the slip boundary condition and be complete vector fields, which are called hydrodynamic Killing vector fields (HKVF) in this paper. Our purpose is to determine surfaces admitting a HKVF. We prove that any connected, orientable surface admitting an HKVF is conformally equivalent to one of the 14 canonical Riemann surfaces, each with either a rotationally or translationally symmetric metric. This paves the way for quantitative investigations of fluid flows associated with Killing vector fields and zonal flows, such as issues of stability and instability, extending its applications potentially to global meteorological phenomena and planetary atmospheric science.

Integrable Generators of Lie Algebras of Vector Fields on textrm{SL}_2(mathbb {C}) and on xy = z^2

For the special linear group textrm{SL}_2(mathbb {C}) and for the singular quadratic Danielewski surface x y = z^2 we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on x y = z^2.

Trajectories of Affine Control Systems and Geodesics of a Spacetime with a Causal Killing Vector Field

We study the geodesic connectedness of a globally hyperbolic spacetime (M,g) admitting a complete smooth Cauchy hypersurface S and endowed with a complete causal Killing vector field K. The main assumptions are that the kernel distribution D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal D$\\end{document} of the one-form induced by K on S is nowhere integrable and that the gradient of g(K,K) is orthogonal to D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {D}$\\end{document}. We approximate the metric g by metrics gε smoothly depending on a real parameter ε and admitting K as a timelike Killing vector field. A known existence result for geodesics of such type of metrics provides a sequence of approximating solutions, joining two given points, of the geodesic equations of (M,g) and whose Lorentzian energy turns out to be bounded thanks to an argument involving trajectories of some affine control systems related with D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {D}$\\end{document}.

Existence of a complete holomorphic vector field via the Kähler–Einstein metric

In this paper, we study the existence of a complete holomorphic vector field on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kahler–Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kahler–Einstein metric whose differential has a constant length. Then, we will construct a complete holomorphic vector field from the gradient vector field of the potential function.

On the resolution of singularities of one-dimensional foliations on three-manifolds

This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.

Differential Equations in a Tangent Category I: Complete Vector Fields, Flows, and Exponentials

This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number object. A curve object is a preinitial object for dynamical systems; dynamical systems may, in turn, be viewed as determining systems of differential equations. The unique map from the curve object to a dynamical system is a solution of the system, and a dynamical system is said to be complete when for all initial conditions there is a solution. A subtle issue concerns the question of when a dynamical system is complete, and the paper provides abstract conditions for this. This abstract formulation also allows new perspectives on topics such as commutative vector fields and flows. In addition, the stronger notion of a differential curve object, which is the centrepiece of the last section of the paper, has exponential maps and forms a differential exponential rig. This rig then, somewhat surprisingly, has an action on every differential object and bundle in the setting. In this manner, in a very strong sense, such a curve object plays the role of the real numbers in standard differential geometry.

Finite groups of diffeomorphisms are topologically determined by a vector field

In a previous work it is shown that every finite group G of diffeomorphisms of a connected smooth manifold M of dimension $$\ge 2$$ equals, up to quotient by the flow, the centralizer of the group of smooth automorphisms of a G-invariant complete vector field X (shortly X describes G). Here the foregoing result is extended to show that every finite group of diffeomorphisms of M is described, within the group of all homeomorphisms of M, by a vector field. As a consequence, it is proved that a finite group of homeomorphisms of a compact connected topological 4-manifold, whose action is free, is described by a continuous flow.

Lorentzian manifolds with causal Killing vector field: causality and geodesic connectedness

We prove that a compact Lorentzian manifold $$({\overline{M}},{\overline{g}})$$ admitting a causal Killing vector field is totally vicious or it contains a compact achronal Killing horizon. In particular a compact spacetime which satisfies the null generic condition and admits a causal Killing vector field is totally vicious. If in addition, its universal Lorentzian covering is globally hyperbolic then it is geodesically connected. In the non-compact case, we prove that a chronological spacetime admitting a complete causal Killing vector field, a smooth spacelike partial Cauchy hypersurface S and satisfying the null generic condition is stably causal. If additionally S is compact then the spacetime is globally hyperbolic. We also determine the geodesic connectedness in this case.

Complete algebraic vector fields on affine surfaces

Let [Formula: see text] be the subgroup of the group [Formula: see text] of holomorphic automorphisms of a normal affine algebraic surface [Formula: see text] generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces [Formula: see text] quasi-homogeneous under [Formula: see text] in terms of the dual graphs of the boundaries [Formula: see text] of their SNC-completions [Formula: see text].

Dense motion propagation from sparse samples

There are many applications for which sparse, or partial sampling of dynamic image data can be used for articulating or estimating motion within the medical imaging area. In this new work, we propose a generalized framework for dense motion propagation from sparse samples which represents an example of transfer learning and manifold alignment, allowing the transfer of knowledge across imaging sources of different domains which exhibit different features. Many such examples exist in medical imaging, from mapping 2D ultrasound or fluoroscopy to 3D or 4D data or monitoring dynamic dose delivery from partial imaging data in radiotherapy. To illustrate this approach we animate, or articulate, a high resolution static MR image with 4D free breathing respiratory motion derived from low resolution sparse planar samples of motion. In this work we demonstrate that sparse sampling of dynamic MRI may be used as a viable approach to successfully build models of free- breathing respiratory motion by constrained articulation. Such models demonstrate high contrast, and high temporal and spatial resolution that have so far been hitherto unavailable with conventional imaging methods. The articulation is based on using a propagation model, in the eigen domain, to estimate complete 4D motion vector fields from sparsely sampled free-breathing dynamic MRI data. We demonstrate that this approach can provide equivalent motion vector fields compared to fully sampled 4D dynamic data, whilst preserving the corresponding high resolution/high contrast inherent in the original static volume. Validation is performed on three 4D MRI datasets using eight extracted slices from a fast 4D acquisition (0.7 s per volume). The estimated deformation fields from sparse sampling are compared to the fully sampled equivalents, resulting in an rms error of the order of 2 mm across the entire image volume. We also present exemplar 4D high contrast, high resolution articulated volunteer datasets using this methodology. This approach facilitates greater freedom in the acquisition of free breathing respiratory motion sequences which may be used to inform motion modelling methods in a range of imaging modalities and demonstrates that sparse sampling of free breathing data may be used within a manifold alignment and transfer learning paradigm to estimate fully sampled motion. The method may also be applied to other examples of sparse sampling to produce dense motion propagation.

A Global Geometric Decomposition of Vector Fields and Applications to Topological Conjugacy

We give a global geometric decomposition of continuously differentiable vector fields on $\mathbb{R}^n$. More precisely, given a vector field of class $\mathcal{C}^{1}$ on $\mathbb{R}^{n}$, and a geometric structure on $\mathbb{R}^n$, we provide a unique global decomposition of the vector field as the sum of a left (right) gradient--like vector field (naturally associated to the geometric structure) with potential function vanishing at the origin, and a vector field which is left (right) orthogonal to the identity, with respect to the geometric structure. As application, we provide a criterion to decide topological conjugacy of complete vector fields of class $\mathcal{C}^1$ on $\mathbb{R}^{n}$ based on topological conjugacy of the corresponding parts given by the associated geometric decompositions.

Local incompatibility of the microlocal spectrum condition with the KMS property along spacelike directions in quantum field theory on curved spacetime

States of a generic quantum field theory on a curved spacetime are considered which satisfy the KMS condition with respect to an evolution associated with a complete (Killing) vector field. It is shown that at any point where the vector field is spacelike, such states cannot satisfy a certain microlocal condition which is weaker than the microlocal spectrum condition in the case of asymptotically free fields.

Translating solitons in Riemannian products

In this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product (R×P,dt2+g0) and the parallel field is X=∂t. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in R×P can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in R×P. When g0 is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results.

Symmetry analysis and some new exact solutions of Born–Infeld equation

This paper propounds the Lie group analysis method for finding exact solutions of Born–Infeld (BI) equation arising in nonlinear electrodynamics. We obtain generators of infinitesimal transformations, commutator table of Lie algebra, the complete geometric vector field, group symmetries and reduction equations. For the set of geometric vector field, we find an optimal system of the vector fields. Each element in this system helps to reduce the main equation into an ordinary differential equation, which provides analytical solution to the BI equation. We perform numerical simulation to obtain an appropriate visual appearance and dynamic behavior of the traced solutions. The nature of the solutions is investigated both analytically and physically through their evolutionary profile by considering appropriate choices of arbitrary constants.

Biharmonic constant mean curvature surfaces in Killing submersions

A $3$-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part, we also classify the proper biharmonic Hopf cylinders in a Killing submersion.

Smooth torus actions are described by a single vector field

Consider a smooth effective action of a torus \mathbb T^n on a connected C^{\infty} -manifold M . Assume that M is not a torus endowed with the natural action. Then we prove that there exists a complete vector field X on M such that the automorphism group of X equals \mathbb T^n \times \mathbb R , where the factor \mathbb R comes from the flow of X and \mathbb T^n is regarded as a subgroup of Diff (M) . Thus one may reconstruct the whole action of \mathbb T^n from a single vector field.

Exceptional values of holomorphic functions. Remarks on a Nishino's Theorem

Let F(z,w)∈O(Cn+1), where (z,w)∈Cn×C. Let z′∈Cn such that F(z′,w) is not constant. If F(z′,w) is not surjective it takes all the values of C minus one π(z′) (Picard). T. Nishino studied in [8]π(z) when n=1, F(z,w) is of finite order in w and π(z) is defined in a set E⊂C with at least one accumulation point. In this work, we see that his result allows to obtain an explicit expression of such a F(z,w) when n≥1 and F(z′,w) is not a constant for any z′∈Cn, and conclude that π(z)=η(z)−1/ξ(z) for η(z) and ξ(z)∈O(Cn) when π(z) is defined on a nonempty open set U⊂Cn. Moreover, we give several applications of this fact. We show that the complement of the graph of π(z) in Cn+1 is dominated by Cn+1 via a family of surjective fiber-preserving holomorphic maps with non-vanishing Jacobian determinant, which are described in terms of the flow of a complete vector field of type C⁎. In particular, Buzzard and Lu's results in [2] applied to π(z) for n=1 can be extended for n≥2. It will allow to define new examples of Oka manifolds.

Structure and equivalence of a class of tube domains with solvable groups of automorphisms

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of ${\frak {g}}(T_{\Omega})$ plays an important role, where ${\frak {g}}(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss in this paper the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.