Generic orbits of the diffeomorphism group of a two-manifold in the space \({\mathfrak G}_{reg}^*\) of regular momenta. (English) Zbl 0751.58003

Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 171-178 (1991).
[For the entire collection see Zbl 0742.00067.]
The author investigates the structure of generic orbits of the diffeomorphism group \(\text{Diff}(M^ 2)\) of a 2-dimensional compact smooth orientable manifold \(M^ 2\) in the space \({\mathfrak G}^*_{reg}\) of regular momenta. It is shown that the existence of orbits of finite codimension in \({\mathfrak G}^*_{reg}\) depends on the topology of \(M^ 2\). In particular, the group \(\text{Diff}(S^ 2)\) has no orbits of finite codimension in \({\mathfrak G}^*_{reg}\). Moreover, even when orbits of finite codimension exist, e.g. for the 2-torus \(T^ 2\), they are far from being typical. This is therefore, quite a contrast to the 1- dimensional case [cf. A. A. Kirillov, Lect. Notes Math. 970, 101- 123 (1982; Zbl 0498.22017)]. The author also obtains global functional moduli for momenta under the coadjoint action of \(\text{Diff}(M^ 2)\).


58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
58D27 Moduli problems for differential geometric structures