CONFORMAL DERIVATIVE AND CONFORMAL TRANSPORTS OVER $\bm{({\bar L}_n,g)}$-SPACES

Transports preserving the angle between two contravariant vector fields but changing their lengths proportional to their own lengths are introduced as "conformal" transports and investigated over [Formula: see text]-spaces. They are more general than the Fermi–Walker transports. In an analogous way as in the case of Fermi–Walker transports a conformal covariant differential operator and its conformal derivative are defined and considered over [Formula: see text]-spaces. Different special types of conformal transports are determined inducing also Fermi–Walker transports for orthogonal vector fields as special cases. Conditions under which the length of a non-null contravariant vector field could swing as a homogeneous harmonic oscillator are established. The results obtained regardless of any concrete field (gravitational) theory could have direct applications in such types of theories.

Download Full-text

Tangent bundle geometry from dynamics: Application to the Kepler problem

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817500475 ◽

2017 ◽

Vol 14 (03) ◽

pp. 1750047 ◽

Cited By ~ 1

Author(s):

J. F. Cariñena ◽

J. Clemente-Gallardo ◽

J. A. Jover-Galtier ◽

G. Marmo

Keyword(s):

Vector Field ◽

Harmonic Oscillator ◽

Tangent Bundle ◽

Vector Fields ◽

Kepler Problem ◽

Second Order ◽

Affine Spaces ◽

Conformal Vector ◽

Conformal Vector Fields

In this paper, we consider a manifold with a dynamical vector field and enquire about the possible tangent bundle structures which would turn the starting vector field into a second-order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second-order and apply the procedure to vector fields conformally related with the harmonic oscillator ([Formula: see text]-oscillators). We select one which covers the vector field describing the Kepler problem.

Download Full-text

Pairings between bounded divergence-measure vector fields and BV functions

Advances in Calculus of Variations ◽

10.1515/acv-2020-0058 ◽

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.

Download Full-text

The geometry of null-like disformal transformations

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819501809 ◽

2019 ◽

Vol 16 (11) ◽

pp. 1950180 ◽

Cited By ~ 2

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.

Download Full-text

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

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000626x ◽

1991 ◽

Vol 11 (3) ◽

pp. 443-454 ◽

Cited By ~ 35

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.

Download Full-text

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

Communications in Contemporary Mathematics ◽

10.1142/s0219199711004233 ◽

2011 ◽

Vol 13 (02) ◽

pp. 191-211 ◽

Cited By ~ 1

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.

Download Full-text

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127495000697 ◽

1995 ◽

Vol 05 (03) ◽

pp. 895-899 ◽

Cited By ~ 1

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.

Download Full-text

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-62-2021-53-66 ◽

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.

Download Full-text

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988781550111x ◽

2015 ◽

Vol 12 (10) ◽

pp. 1550111 ◽

Cited By ~ 1

Author(s):

Mircea Crasmareanu ◽

Camelia Frigioiu

Keyword(s):

Vector Field ◽

Vector Fields ◽

Killing Vector ◽

Three Dimensional ◽

Ricci Soliton ◽

Conformal Killing ◽

Space Forms ◽

Potential Vector ◽

Legendre Curves ◽

Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

Download Full-text

Flows of measures generated by vector fields

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000312 ◽

2018 ◽

Vol 148 (4) ◽

pp. 773-818

Author(s):

Emanuele Paolini ◽

Eugene Stepanov

Keyword(s):

Vector Field ◽

Vector Fields ◽

Borel Measure ◽

Weak Sense ◽

Positive Borel Measure ◽

Smooth Structure ◽

Smooth Vector ◽

Integral Curves ◽

The Given ◽

Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

Download Full-text

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

Advances in Geometry ◽

10.1515/advgeom-2017-0014 ◽

2018 ◽

Vol 18 (3) ◽

pp. 337-344 ◽

Cited By ~ 1

Author(s):

Ju Tan ◽

Shaoqiang Deng

Keyword(s):

Vector Field ◽

Lie Groups ◽

Lie Algebras ◽

Critical Points ◽

Vector Fields ◽

Energy Functional ◽

Smooth Vector ◽

Solvable Lie Groups ◽

Invariant Vector ◽

Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

Download Full-text