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Charles Ehresmann's concepts in differential geometry

Paulette Libermann (2007)

Banach Center Publications

We outline some of the tools C. Ehresmann introduced in Differential Geometry (fiber bundles, connections, jets, groupoids, pseudogroups). We emphasize two aspects of C. Ehresmann's works: use of Cartan notations for the theory of connections and semi-holonomic jets.

Classification of Nash manifolds

Masahiro Shiota (1983)

Annales de l'institut Fourier

A semi-algebraic analytic manifold and a semi-algebraic analytic map are called a Nash manifold and a Nash map respectively. We clarify the category of Nash manifolds and Nash maps.

Classification of principal connections naturally induced on W 2 P E

Jan Vondra (2008)

Archivum Mathematicum

We consider a vector bundle E → M and the principal bundle P E of frames of E . Let K be a principal connection on P E and let Λ be a linear connection on M . We classify all principal connections on W 2 P E = P 2 M × M J 2 P E naturally given by K and Λ .

Cohomologie de dolbeault le long des feuilles de certains feuilletages complexes

Aziz El Kacimi Alaoui, Jihène Slimène (2010)

Annales de l’institut Fourier

La cohomologie de Dolbeault feuilletée mesure l’obstruction à résoudre le problème de Cauchy-Riemann le long des feuilles d’un feuilletage complexe. En utilisant des méthodes de cohomologie des groupes, nous calculons cette cohomologie pour deux classes de feuilletages : i) le feuilletage complexe affine de Reeb de dimension (complexe) 2 sur la variété de Hopf de dimension 5 ; ii) les feuilletages complexes sur le tore hyperbolique (fibration en tores de dimension n au-dessus d’un cercle et de monodromie...

Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case

Josef Janyška, Martin Markl (2012)

Archivum Mathematicum

This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.

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