On compact Ricci solitons in Finsler geometry

Mohamad Yar Ahmadi; Behroz Bidabad

doi:10.1016/j.crma.2015.09.012

Differential geometry

On compact Ricci solitons in Finsler geometry

Presented by: the Editorial Board

Mohamad Yar Ahmadi ¹; Behroz Bidabad ¹

¹ Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1023-1027.

Abstract
Résumé

Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work, it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group, and hence the first de Rham cohomology group vanishes.

Les solitons de Ricci sur les espaces de Finsler, précédemment définis et étudiés par les auteurs de la présente note, sont une généralisation des espaces d'Einstein, et peuvent être considérés comme des solutions du flot de Ricci sur les variétés finslériennes compactes. Dans ce travail, on démontre qu'un soliton de Ricci complet contractant en temps croissant sur un espace de Finsler est compact si et seulement s'il est borné. En outre, il est démontré qu'un soliton de Ricci contractant compact donne lieu à un groupe fondamental de type fini et donc que le premier groupe de cohomologie s'annule.

Received: 2015-01-23
Accepted: 2015-09-14
Published online: 2015-10-21

DOI: 10.1016/j.crma.2015.09.012

Author's affiliations:

Mohamad Yar Ahmadi ¹; Behroz Bidabad ¹

¹ Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

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     title = {On compact {Ricci} solitons in {Finsler} geometry},
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     pages = {1023--1027},
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     doi = {10.1016/j.crma.2015.09.012},
     language = {en},
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Mohamad Yar Ahmadi; Behroz Bidabad. On compact Ricci solitons in Finsler geometry. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1023-1027. doi : 10.1016/j.crma.2015.09.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.012/

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