Rigidity and concircular symmetries

2021 ◽  
pp. 2150202
Author(s):  
Andrés Franco
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Lorentzian Geometry ◽  
Rigidity Result ◽  
The Relationship

We present a rigidity result in Lorentzian geometry, related to Bartnik’s conjecture, under the hypothesis of the existence of a concircular vector field with certain properties. We also study the relationship between two different notions of concircular vector fields found in the literature, and give conditions for a generalized Robertson–Walker spacetime to have a timelike concircular vector field.

scholarly journals Kähler metrics via Lorentzian Geometry in dimension four

2019 ◽  
Vol 7 (1) ◽  
pp. 36-61 ◽  
Author(s):  
Amir Babak Aazami ◽  
Gideon Maschler
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Plane Waves ◽  
Killing Vector ◽  
Lorentzian Geometry ◽  
De Sitter ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Open Set ◽  
Sitter Spacetime

AbstractGiven a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.

Uniqueness and non-uniqueness of normal forms for vector fields

1988 ◽  
Vol 108 (1-2) ◽  
pp. 27-33 ◽  
Author(s):  
Alberto Baider ◽  
Richard Churchill
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Resonance Problem ◽  
Main Result Theorem ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Result Theorem ◽  
The Relationship

SynopsisThe use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.

A new approach to the bienergy and biangle of a moving particle lying in a surface of lorentzian space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Talat Körpınar ◽  
Yasin Ünlütürk
Keyword(s):  
Vector Fields ◽  
Elastic Surface ◽  
Geometrical Characterization ◽  
Lorentzian Space ◽  
New Approach ◽  
Darboux Vector ◽  
Moving Particle ◽  
The Relationship ◽  
Surface Curve

AbstractIn this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

scholarly journals Pairings between bounded divergence-measure vector fields and BV functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Virginia De Cicco ◽  
Annalisa Malusa
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Vector Fields ◽  
Hyperbolic Conservation Laws ◽  
Divergence Measure ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Bv Functions ◽  
Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

scholarly journals The geometry of null-like disformal transformations

2019 ◽  
Vol 16 (11) ◽  
pp. 1950180 ◽  
Author(s):  
I. P. Lobo ◽  
G. G. Carvalho
Keyword(s):  
Vector Field ◽  
Conformal Transformation ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Spinor Structure ◽  
Schwarzschild Metric ◽  
Killing Vectors ◽  
Symmetry Properties ◽  
Geometric Objects

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

Some Properties of Symplectic Manifolds S on the Projective Tangent Bundle PTM

2011 ◽  
Vol 187 ◽  
pp. 483-486
Author(s):  
Yong He ◽  
Xiao Ying Lu ◽  
Wei Na Lu
Keyword(s):  
Vector Field ◽  
Symplectic Manifold ◽  
Tangent Bundle ◽  
Fiber Bundle ◽  
Symplectic Manifolds ◽  
Lie Derivative ◽  
The Relationship

In this paper, we show the relationship between 2-form of the two projective tangent bundle and the relationship between 2-form on projective tangent bundle and 1-form on by using the theory of fiber bundle and the properties of symplectic manifold of the projective tangent bundle . Moreover, we derived a simpler formula of Lie derivative of a special vector field, which is on the projective tangent bundle.

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

SINGULAR CYCLES AND Ck-ROBUST TRANSITIVE SET ON MANIFOLD WITH BOUNDARY

2011 ◽  
Vol 13 (02) ◽  
pp. 191-211 ◽  
Author(s):  
D. CARRASCO-OLIVERA ◽  
C. A. MORALES ◽  
B. SAN MARTÍN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Manifold With Boundary ◽  
Transitive Set ◽  
Singular Cycle ◽  
Hyperbolic Equilibrium

Let M be a 3-manifold with boundary ∂M. Let X be a C∞, vector field on M, tangent to ∂M, exhibiting a singular cycle associated to a hyperbolic equilibrium σ∈∂M with real eigenvalues λss < λs < 0 < λu satisfying λs - λss - 2λu > 0. We prove under generic conditions and k large enough the existence of a Ck robust transitive set of X, that is, any Ck vector field Ck close to X exhibits a transitive set containing the cycle. In particular, C∞ vector fields exhibiting Ck robust transitive sets, for k large enough, which are not singular-hyperbolic do exist on any compact 3-manifold with boundary.

FINDING PARAMETERS FOR CHUA’S OSCILLATOR WHICH IS TOPOLOGICALLY CONJUGATE TO SYSTEMS WITH A SCALAR NONLINEARITY

1995 ◽  
Vol 05 (03) ◽  
pp. 895-899 ◽  
Author(s):  
CHAI WAH WU ◽  
LEON O. CHUA
Keyword(s):  
Vector Field ◽  
Large Class ◽  
Vector Fields ◽  
Colpitts Oscillator ◽  
Chua’S Oscillator ◽  
Scalar Nonlinearity

Chua’s oscillator is topologically conjugate to a large class of vector fields with a scalar non-linearity. In this letter, we give an algorithm which, given a vector field in this class, finds the parameters for Chua’s oscillator for which Chua’s oscillator is topologically conjugate to it. We illustrate this by transforming Sparrow’s system and the chaotic Colpitts oscillator into equivalent Chua’s oscillators.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

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