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.
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.
Revised:
Accepted:
Published online:
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
@article{AIF_2017__67_2_843_0, author = {Matveev, Vladimir S. and Nikolayevsky, Yuri}, title = {Locally conformally {Berwald} manifolds and compact quotients of reducible manifolds by homotheties}, journal = {Annales de l'Institut Fourier}, pages = {843--862}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {67}, number = {2}, year = {2017}, doi = {10.5802/aif.3097}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3097/} }
TY - JOUR AU - Matveev, Vladimir S. AU - Nikolayevsky, Yuri TI - Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties JO - Annales de l'Institut Fourier PY - 2017 SP - 843 EP - 862 VL - 67 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3097/ DO - 10.5802/aif.3097 LA - en ID - AIF_2017__67_2_843_0 ER -
%0 Journal Article %A Matveev, Vladimir S. %A Nikolayevsky, Yuri %T Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties %J Annales de l'Institut Fourier %D 2017 %P 843-862 %V 67 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3097/ %R 10.5802/aif.3097 %G en %F AIF_2017__67_2_843_0
Matveev, Vladimir S.; Nikolayevsky, Yuri. Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 843-862. doi : 10.5802/aif.3097. https://aif.centre-mersenne.org/articles/10.5802/aif.3097/
[1] On the irreducibility of locally metric connections, J. Reine Angew. Math., Volume 714 (2016), pp. 123-150 | DOI
[2] Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France, Volume 83 (1955), pp. 279-330 | DOI
[3] Volume forms in Finsler spaces, Houston J. Math., Volume 25 (1999) no. 4, pp. 625-640
[4] Closed similarity manifolds, Comment. Math. Helv., Volume 55 (1980) no. 4, pp. 576-582 | DOI
[5] Similarity structures and de Rham decomposition (2016) (http://arxiv.org/abs/1507.05573v2)
[6] A counterexample to Belgun-Moroianu conjecture, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 5, pp. 455-457 | DOI
[7] The Binet-Legendre metric in Finsler geometry, Geom. Topol., Volume 16 (2012) no. 4, pp. 2135-2170 | DOI
[8] Géométrie de Finsler sur les espaces symétriques, Université de Genève, Switzerland (1995) (Ph. D. Thesis)
[9] Structures de Finsler invariantes sur les espaces symétriques, C. R. Acad. Sci. Paris Sér. I Math., Volume 321 (1995) no. 11, pp. 1455-1458
[10] Twisted products in pseudo-Riemannian geometry, Geom. Dedicata, Volume 48 (1993) no. 1, pp. 15-25 | DOI
[11] On the transitivity of holonomy systems, Ann. Math., Volume 76 (1962), pp. 213-234 | DOI
[12] Positive definite Berwald spaces. Structure theorems on Berwald spaces, Tensor, Volume 35 (1981) no. 1, pp. 25-39
[13] A new proof of Szabó’s theorem on the Riemann-metrizability of Berwald manifolds, Acta Math. Acad. Paedagog. Nyházi., Volume 21 (2005) no. 2, p. 199-204 (electronic)
[14] On a scale function for testing the conformality of a Finsler manifold to a Berwald manifold, J. Geom. Phys., Volume 54 (2005) no. 4, pp. 454-475 | DOI
[15] On geometric vector fields of Minkowski spaces and their applications, Differential Geom. Appl., Volume 24 (2006) no. 1, pp. 1-20 | DOI
Cited by Sources: