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Schwarzian derivative related to modules of differential operators on a locally projective manifold

S. Bouarroudj, V. Ovsienko (2000)

Banach Center Publications

We introduce a 1-cocycle on the group of diffeomorphisms Diff(M) of a smooth manifold M endowed with a projective connection. This cocycle represents a nontrivial cohomology class of Diff(M) related to the Diff(M)-modules of second order linear differential operators on M. In the one-dimensional case, this cocycle coincides with the Schwarzian derivative, while, in the multi-dimensional case, it represents its natural and new generalization. This work is a continuation of [3] where the same problems...

Second cohomology classes of the group of C 1 -flat diffeomorphisms

Tomohiko Ishida (2012)

Annales de l’institut Fourier

We study the cohomology of the group consisting of all C ∞ -diffeomorphisms of the line, which are C 1 -flat to the identity at the origin. We construct non-trivial two second real cohomology classes and uncountably many second integral homology classes of this group.

Singularities and normal forms of generic 2-distributions on 3-manifolds

B. Jakubczyk, M. Zhitomirskiĭ (1995)

Studia Mathematica

We give a complete classification of germs of generic 2-distributions on 3-manifolds. By a 2-distribution we mean either a module generated by two vector fields (at singular points its dimension decreases) or a Pfaff equation, i.e. a module generated by a differential 1-form (at singular points the dimension of its kernel increases).

Some critical almost Kähler structures

Takashi Oguro, Kouei Sekigawa (2008)

Colloquium Mathematicae

We consider the set of all almost Kähler structures (g,J) on a 2n-dimensional compact orientable manifold M and study a critical point of the functional ℱ λ , μ ( J , g ) = ∫ M ( λ τ + μ τ * ) d M g with respect to the scalar curvature τ and the *-scalar curvature τ*. We show that an almost Kähler structure (J,g) is a critical point of ℱ - 1 , 1 if and only if (J,g) is a Kähler structure on M.

Spaces of measurable functions

Piotr Niemiec (2013)

Open Mathematics

For a metrizable space X and a finite measure space (Ω, 𝔐 , µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of 𝔐 -measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

Spaces of polynomials with roots of bounded multiplicity

M. Guest, A. Kozlowski, K. Yamaguchi (1999)

Fundamenta Mathematicae

We describe an alternative approach to some results of Vassiliev ([Va1]) on spaces of polynomials, by applying the "scanning method" used by Segal ([Se2]) in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.

Spaces of upper semicontinuous multi-valued functions on complete metric spaces

Katsuro Sakai, Shigenori Uehara (1999)

Fundamenta Mathematicae

Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by U S C C B ( X ) the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify φ ∈ U S C C B ( X ) with its graph which is a closed subset of X × ℝ. The space U S C C B ( X ) admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then U S C C B ( X ) is homeomorphic to a...

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