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One-dimensional infinitesimal-birational duality through differential operators

Tomasz Maszczyk (2006)

Fundamenta Mathematicae

The structure of filtered algebras of Grothendieck's differential operators on a smooth fat point in a curve and graded Poisson algebras of their principal symbols is explicitly determined. A related infinitesimal-birational duality realized by a Springer type resolution of singularities and the Fourier transformation is presented. This algebro-geometrical duality is quantized in appropriate sense and its quantum origin is explained.

Only one of generalized gradients can be elliptic

Jerzy Kalina, Antoni Pierzchalski, Paweł Walczak (1997)

Annales Polonici Mathematici

Decomposing the space of k-tensors on a manifold M into the components invariant and irreducible under the action of GL(n) (or O(n) when M carries a Riemannian structure) one can define generalized gradients as differential operators obtained from a linear connection ∇ on M by restriction and projection to such components. We study the ellipticity of gradients defined in this way.

Open book structures and unicity of minimal submanifolds

R. Hardt, Harold Rosenberg (1990)

Annales de l'institut Fourier

We prove unicity of certain minimal submanifolds, for example Clifford annuli in S 3 . The idea is to consider the placement of the submanifold with respect to the (singular) foliation of S 3 by the Clifford annuli whose boundary are two fixed great circles a distance π / 2 apart.

Open books on contact five-manifolds

Otto van Koert (2008)

Annales de l’institut Fourier

By using open book techniques we give an alternative proof of a theorem about the existence of contact structures on five-manifolds due to Geiges. The theorem asserts that simply-connected five-manifolds admit a contact structure in every homotopy class of almost contact structures.

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