Foliated vector bundles and Riemannian foliations

Paul Popescu; Marcela Popescu

doi:10.1016/j.crma.2011.02.017

Differential Geometry

Foliated vector bundles and Riemannian foliations

Presented by: Étienne Ghys

Paul Popescu ¹; Marcela Popescu ¹

¹ Département de mathematiques appliquées, université de Craiova, rue Al. Cuza, No. 13, 200585 Craiova, Romania

Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 445-449.

Abstract
Résumé

In this Note we prove the equivalence between the Riemannian foliation and each of the following conditions: 1) the lifted foliation $F^{r}$ on the bundle of r-transverse jets is Riemannian for $r ⩾ 1$ ; 2) the foliation $F_{0}^{r}$ on the slashed $J_{0}^{r}$ is Riemannian and vertically exact for $r ⩾ 1$ ; 3) there exists a positively admissible transverse Lagrangian on $J_{0}^{r} E$ , the r-transverse slashed jet bundle of a foliated bundle $E \to M$ , for $r ⩾ 1$ .

Dans cette Note on établie lʼéquivanence entre la propriété pour un feuilletage dʼêtre riemannien et chacune des conditions suivantes : 1) le feuilletage rélevé $F^{r}$ sur lʼespace des jets r-transverses est riemannien pour une certaine valeur de $r ⩾ 1$ ; 2) le feuilletage rélevé $F_{0}^{r}$ sur lʼespace réduit des jets r-transverses est riemannien et verticalement exact pour une certaine valeur de $r ⩾ 1$ ; 3) il existe un lagrangien positif, admissible et transvers sur $J_{0}^{r} E$ , le fibré réduit des jets r-transverses dʼun fibré vectoriel $E \to M$ , pour une certaine valeur $r ⩾ 1$ .

Received: 2010-02-14
Accepted: 2011-02-17
Published online: 2011-03-04

DOI: 10.1016/j.crma.2011.02.017

Author's affiliations:

Paul Popescu ¹; Marcela Popescu ¹

¹ Département de mathematiques appliquées, université de Craiova, rue Al. Cuza, No. 13, 200585 Craiova, Romania

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     author = {Paul Popescu and Marcela Popescu},
     title = {Foliated vector bundles and {Riemannian} foliations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {445--449},
     publisher = {Elsevier},
     volume = {349},
     number = {7-8},
     year = {2011},
     doi = {10.1016/j.crma.2011.02.017},
     language = {en},
}

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%A Marcela Popescu
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Paul Popescu; Marcela Popescu. Foliated vector bundles and Riemannian foliations. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 445-449. doi : 10.1016/j.crma.2011.02.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.017/

References
Cited by

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