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A convex Darboux theorem

Pierre-André Chiappori, Ivar Ekeland (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

A generalization of the exterior product of differential forms combining Hom-valued forms

Christian Gross (1997)

Commentationes Mathematicae Universitatis Carolinae

This article deals with vector valued differential forms on C -manifolds. As a generalization of the exterior product, we introduce an operator that combines Hom ( s ( W ) , Z ) -valued forms with Hom ( s ( V ) , W ) -valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.

Curvature and the equivalence problem in sub-Riemannian geometry

Erlend Grong (2022)

Archivum Mathematicum

These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which...

Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

Jun-Muk Hwang (2010)

Annales scientifiques de l'École Normale Supérieure

We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety Z , a family of minimal rational curves with Z -isotrivial varieties of minimal rational tangents...

Espaces variationnels et mécanique

Joseph Klein (1962)

Annales de l'institut Fourier

Ce travail est essentiellement consacré aux systèmes dynamiques Σ non conservatifs, la force généralisée dépendant à la fois des paramètres de position x α et de vitesse y α . V désignant l’espace-temps de configuration, V l’espace fibré des vecteurs tangents, W celui des directions tangentes à V , on caractérise Σ par son lagrangien homogène L et le tenseur-force S antisymétrique dont le produit contracté par le vecteur vitesse donne le vecteur force généralisé.Dans la première partie, on étudie l’algèbre...

Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere

Joël Merker, Masoud Sabzevari (2012)

Open Mathematics

We study effectively the Cartan geometry of Levi-nondegenerate C 6-smooth hypersurfaces M 3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M 3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.

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