arXiv subjects
Publications and source records attributed to V. Ovsienko.
We discuss Ghys' theorem on 4 zeroes of the Schwarzian derivative and its relation with flattening points of Legendrian curves and Sturm theory.
The space of linear differential operators on a smooth manifold $M$ has a natural one-parameter family of $Diff(M)$ (and $Vect(M)$)-module structures, defined by their action on the space of tensor-densities. It is shown that, in the case of second order differential operators, the $Vect(M)$-module structures are equivalent for any degree of tensor-densities except for three critical values: $\{0,{1\over 2},1\}$. Second order analogue of the Lie derivative appears as an intertwining operator between the spaces of second order differential operators on tensor-densities.
We classify non-trivial (non-central) extensions of the group $Diff^+(S^1)$ of all diffeomorphisms of the circle preserving its orientation and of the Lie algebra $Vect (S^1)$ of vector fields on $S^1$, by the modules of tensor-densities on $S^1$. The result is: 4 non-trivial extensions of $Diff^+(S^1)$ and 7 non-trivial extensions of $Vect (S^1)$. Analogous results hold for the Virasoro group and the Virasoro algebra. We also classify central extensions of constructed Lie algebras. CPT-94/P.3024