# Geometry of integrable vector fields on surfaces

Updated: 04/April/2012

This paper is on schedule. It is almost finished now, and has about 30 pages. To be put on arxiv tomorrow.

Updated: 19/03/2012

Section: Generic nilpotent singularities

(X,F) where F regular, X nilpotent:

F =F(y),

X = y d/dx + …

Since X(F) = 0 –> X = (y + …) d/dx

Normal forms: Takens (-Bogdanov), Gong in the integrable case

Assume y + … is generic in the expression of X

2 ways to write:

a) As Takens-Gong –> X = (y + g(x) ) d/dx where g(x) = ax^2 + odd_function (x), and assume that a \neq 0.

b) Normalize the set {X=0} (which is a parabol) –>

X = h(x,y). (y – x^2/2) d/dx

where h(x,y) is a smooth function, h(0) \neq 0

Question: which way is better to detect local invariants ?

The eigenvalue function on the curve S= {X= 0}

Using the second expression X = h(x,y). (y – x^2/2) d/dx,

we have E (q) = – x(q) h(q) for each q \in S ={X=0} = {y = x^2}

where E(q) means eigenvalue at q.

Question: up to equivalence, the behavior of the function h(x,y) is only important at the curve S ?

—-

This is another work in progress with my PhD student Minh. After finishing our paper on the geometry of Rn actions on n-manifolds, we now have time to write up this one.

To finish before: 15/April/20012

Here we want to study the properties and obtain a classification (up to continuous or smooth equivalence) of integrable systems of type (1,1), i.e. just 1 vector field and 1 first integral on a 2-dimensional surface.

0) Different equvalences to consider:

– local / semilocal / global

– continuous / smooth

– geometric (can multiply the vector field by a function)  / exact ?

I) Local structure of singularities

– Nondegenerate / Morse ? Geometric linearization ? Smooth classification ? Poincare-Dulac normal form ? Problem of periods of the center ?

– Algebraically isolated singularities

– Singularities of the function ?

Remark Our singularities are singularities of the couple (X,f), and not of X alone.

* The function is regular: 1-dim family of hyperbolic vector fields in dim 1.

* The function is Morse: elliptic and hyperbolic

* The vector field is regular, but the function has a singularity (Morse-Bott function)

* Isolated singular point of the function, the vectoe field also vanishes at that point. Hamiltonianization of this situation ?

* Degenerate centers ?

* Multiplicity of the function ? (Important for the global picture)

II) Semi-local classification

– Hamiltonianization

– Regularized period functions

III) Global picture

– Global invarinats ?

* Marked singularity graph

* Monodormy ?! On each invariant curve ! (similar to our paper on Rn actions)

IV)  Related works / References

– J Llibre & co ?

– Kudryavtseva ?

* Colin & Vey, Lemme de Morse isochore, 1979 ?

* Jaume Llibre, Clàudia Valls: Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree, Advances in Mathematics (22 February 2011)

* Sergiy Maksymenko, Functions with isolated singularities on surfaces, arxiv 0806.4704

*Dufour Molino Toulet, Classification des systèmes intégrables en dimension 2 et invariants des modèles de Fomenko, CRAS, 1994

* Xianghong Gong, Annales Fourier 1995 (?) Integrable nilpotent case.

* Oshemkov Morse functions on two-dimensional surfaces. Encoding of singularities 1994

*  A A Oshemkov and V V Sharko   Classification of Morse-Smale flows on two-dimensional manifolds, 1998 Sb. Math. 189 1205

* Takens: Singularities of vector fields, Publ IHES 1974 (for the nipoent singularity)

* NT Zung: Nondegenerate singuarities … (2011, revised 2012)

* NT Zung: Linearization of smooth completely integrable vector fields (in preparation, 2012)

* NT Zung & NV Minh: Geometry of Rn-actions on n-manifolds, 2012

——

We’ll submit this paper to a special issue in Acta Math Vietnam