no. 2
Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties
[Variétés localement conformément Berwaldiennes et quotients compacts de variétés réductibles par d’homothéthies]
Matveev, Vladimir S.1 ; Nikolayevsky, Yuri2
1 Institute of Mathematics Friedrich-Schiller-Universität Jena 07737 Jena (Germany)
2 Department of Mathematics and Statistics La Trobe University Melbourne, Victoria, 3086 (Australia)
Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 843-862.

Nous étudions des métriques qui sont localement, mais pas globalement conformément Berwaldiennes. Nous démontrons que la caractérisation de telles métriques est équivalente à la caractérisation des variétés Riemanniennes incomplètes et simplement connexes qui ont un groupe d’holonomie réductible tel que le quotient par un groupe d’homothéthies est fermé. De plus, nous démontrons un théorème de décomposition du type de Rham disant que si une telle variété est analytique, elle est isométrique à un produit Riemannien d’un espace Euclidien et d’une variété incomplète.

We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is closed. We further prove a de Rham type splitting theorem which states that if such a manifold is analytic, it is isometric to the Riemannian product of a Euclidean space and an incomplete manifold.

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DOI : 10.5802/aif.3097
Classification : 53C60, 53C22, 53B40, 53C29
Keywords: Finsler manifold, Berwald manifold, homothety group, reducible holonomy
Mot clés : Variété finslerienne, variété Berwaldienne, groupe d’homothéthies, holonomie réductible
Matveev, Vladimir S. 1 ; Nikolayevsky, Yuri 2

1 Institute of Mathematics Friedrich-Schiller-Universität Jena 07737 Jena (Germany)
2 Department of Mathematics and Statistics La Trobe University Melbourne, Victoria, 3086 (Australia) 
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Matveev, Vladimir S.; Nikolayevsky, Yuri. Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties. Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 843-862. doi : 10.5802/aif.3097. http://archive.numdam.org/articles/10.5802/aif.3097/

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