Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(6): 2119-2139

Online first article August 10, 2017      Printed November 30, 2017

https://doi.org/10.4134/BKMS.b160780

Copyright © The Korean Mathematical Society.

Deformation of Cartan curvature on Finsler manifolds

Behroz Bidabad, Alireza Shahi, Mohamad Yar Ahmadi

Amirkabir University of Technology, Amirkabir University of Technology, Amirkabir University of Technology

Abstract

Here, certain Ricci flow for Finsler $n$-manifolds is considered and deformation of Cartan $hh$-curvature, as well as Ricci tensor and scalar curvature, are derived for spaces of scalar flag curvature. As an application, it is shown that on a family of Finsler manifolds of constant flag curvature, the scalar curvature satisfies the so-called heat-type equation. Hence on a compact Finsler manifold of constant flag curvature of initial non-negative scalar curvature, the scalar curvature remains non-negative by Ricci flow and blows up in a short time.

Keywords: deformation, Finsler Ricci flow, blow up, evolution, heat-type equation

MSC numbers: 53C60, 53C44