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Local reduction theorems and invariants for singular contact structures

Bronislaw Jakubczyk, Michail Zhitomirskii (2001)

Annales de l’institut Fourier

A differential 1-form on a ( 2 k + 1 ) -dimensional manifolds M defines a singular contact structure if the set S of points where the contact condition is not satisfied, S = { p M : ( ω ( d ω ) k ( p ) = 0 } , is nowhere dense in M . Then S is a hypersurface with singularities and the restriction of ω to S can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation ( ω ) generated by ω is determined, up to a diffeomorphism, by its restriction to S , if we eliminate certain degenerated singularities...

Local structural stability of C 2 integrable 1-forms

Alcides Lins Neto (1977)

Annales de l'institut Fourier

In this work we consider a class of germs of singularities of integrable 1-forms in R n which are structurally stable in class C r ( r 2 if n = 3 , r 4 if n 4 ), whose 1-jet is zero at the singularity. In this class the stability depends essentially on the fact that the perturbations allowed are integrable.

Local symplectic algebra of quasi-homogeneous curves

Wojciech Domitrz (2009)

Fundamenta Mathematicae

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of a 𝕂-analytic curve is a finite-dimensional vector space. We also show that the action of local diffeomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the infinitesimal action of...

Locally variational invariant field equations and global currents: Chern-Simons theories

Mauro Francaviglia, M. Palese, E. Winterroth (2012)

Communications in Mathematics

We introduce the concept of conserved current variationally associated with locally variational invariant field equations. The invariance of the variation of the corresponding local presentation is a sufficient condition for the current beeing variationally equivalent to a global one. The case of a Chern-Simons theory is worked out and a global current is variationally associated with a Chern-Simons local Lagrangian.

Logarithmic structure of the generalized bifurcation set

S. Janeczko (1996)

Annales Polonici Mathematici

Let G : n × r be a holomorphic family of functions. If Λ n × r , π r : n × r r is an analytic variety then    Q Λ ( G ) = ( x , u ) n × r : G ( · , u ) h a s a c r i t i c a l p o i n t i n Λ π r - 1 ( u ) is a natural generalization of the bifurcation variety of G. We investigate the local structure of Q Λ ( G ) for locally trivial deformations of Λ = π r - 1 ( 0 ) . In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.

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