This is the research project of a PhD student of mine. Actually the general project is sufficiently large for a number of PhD theses. There are lots of open questions in the non-Hamiltonian case.
What my student is doing is to study simplest (mostly low-dimensional) cases, and with only nondegenerate singularities:
- Systems of type (1,1), i.e. 1 vector field and 1 function, dimension 2. Already in this case, the picture is non-trivial.
- Any dimension: a real classification of nondegenerate singular points of type (n,0), i.e. zero function and n vector fields
- Dimension 3: local strcture of type (1,2), type (2,1), and also some questions about the global structure.
- Dimension 4: Monodromy phenomenon around certain singularities in dimension 4.
That will be enough for his thesis. He must write up a research article before summer.
Systems of type (n,0) ?
(n,0) means n commuting vector fields and 0 function on a connected manifold M of dimension n. We will assume that the vector fields are complete –> action of Rn on M.
Non-singular case –> Liouville theorem –> M is a torus in the compact case, or a cylinder Rk x Tm (k+m=n) in the non-compact case.
Nondegenerate singularities:
Locally = direct product of hyperbolic components (of dimension 1), and elliptic components (of dimension 2), and non-singular part.
“boundary” of a hyperbolic component (at “infinity”) = a point
“boundary” of an elliptic component = a circle T1 or a point
Global topology question: what manifolds can admit integrable (n,0) systems with only nondegenerate singularities ?
n=2 –> S2, T2, non-orietable RP2 and K2 ?
hyperbolic surfaces can’t admit this thing ?
n=3 –> lense spaces OK, graph-manifolds OK or not ?
any n ? Elliptic manifolds ???