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Of all publications in the section: 5
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Article
Verbitsky M. Differential Geometry and its Application. 2011. Vol. 29. No. 1. P. 101-107.
Article
Izosimov A. Differential Geometry and its Application. 2013. Vol. 31. P. 557-567.

A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.

We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.

Article
Felikson А., Natanzon S. M. Differential Geometry and its Application. 2012. Vol. 30. No. 5. P. 490-508.

We consider (local) parameterizations of Teichmüller space Tg,n (of genus g hyperbolic surfaces with n boundary components) by lengths of 6 g- 6 + 3 n geodesics. We find a large family of suitable sets of 6 g- 6 + 3. n geodesics, each set forming a special structure called "admissible double pants decomposition". For admissible double pants decompositions containing no double curves we show that the lengths of curves contained in the decomposition determine the point of Tg,n up to finitely many choices. Moreover, these lengths provide a local coordinate in a neighborhood of all points of Tg,n{set minus}X where X is a union of 3 g- 3 + n hypersurfaces. Furthermore, there exists a groupoid acting transitively on admissible double pants decompositions and generated by transformations exchanging only one curve of the decomposition. The local charts arising from different double pants decompositions compose an atlas covering the Teichmüller space. The gluings of the adjacent charts are coming from the elementary transformations of the decompositions, the gluing functions are algebraic. The same charts provide an atlas for a large part of the boundary strata in Deligne-Mumford compactification of the moduli space Mg,n.

We suggest a general method of computation of the homology of certain smooth covers $\widehat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces of algebraic curves with level $m$ structures. The method is based on the lifting of the Strebel-Penner stratification of $\mathcal{M}_{g,1}\br{\mathbb{C}}$. We apply this method for $g\leq 2$ and obtain Betti numbers; these results are consistent with Penner and Harer-Zagier results on Euler characteristics.