Bicrossed products induced by Poisson vector fields and their integrability

2016 ◽  
Vol 13 (03) ◽  
pp. 1650022
Author(s):  
Samson Apourewagne Djiba ◽  
Aïssa Wade
Keyword(s):  
Vector Field ◽  
Cohomology Class ◽  
Vector Fields ◽  
Poisson Manifold ◽  
Lie Algebroid ◽  
Jet Bundle

First we show that, associated to any Poisson vector field [Formula: see text] on a Poisson manifold [Formula: see text], there is a canonical Lie algebroid structure on the first jet bundle [Formula: see text] which, depends only on the cohomology class of [Formula: see text]. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and [Formula: see text]-algebras.

scholarly journals On Sha's Secondary Chern–Euler Class

2012 ◽  
Vol 55 (3) ◽  
pp. 586-596
Author(s):  
Zhaohu Nie
Keyword(s):  
Vector Field ◽  
Cohomology Class ◽  
Vector Fields ◽  
Euler Class ◽  
Boundary Term ◽  
Manifold With Boundary ◽  
Normal Vectors ◽  
Tangential Projection ◽  
Euler Form ◽  
Hopf Theorem

AbstractFor a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

INTEGRABILITY OF PLANE FIELDS DEFINED BY 2-VECTOR FIELDS

2005 ◽  
Vol 16 (02) ◽  
pp. 197-212 ◽  
Author(s):  
KENTARO MIKAMI ◽  
TADAYOSHI MIZUTANI
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Complete Integrability ◽  
Lie Algebroids ◽  
Lie Algebroid

Given a 2-vector field on a manifold, first we briefly discuss the complete integrability of the distribution which is the image of the 2-vector field. Then we show that a new Lie algebroid is defined on such a maniold which is coincident with the cotangent Lie algebroid when the 2-vector field is Poisson. The result is extended to the case of Lie algebroids.

L∞-actions of Lie algebroids

2021 ◽  
pp. 2150013
Author(s):  
Olivier Brahic ◽  
Marco Zambon
Keyword(s):  
Vector Field ◽  
Direct Product ◽  
Vector Fields ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Geometric Structures ◽  
Courant Algebroids ◽  
Graded Manifold

We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable [Formula: see text]-algebra morphisms. On the “semi-direct product” we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geometric structures can be described by homological vector fields as above, we can display many explicit examples, involving Lie algebroids (including extensions, representations up to homotopy and their cocycles) as well as transitive Courant algebroids.

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.

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.

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

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
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.

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