We study one of the open problems in Finsler geometry presented by Matsumoto-Shimada in 1977, about the existence of a concrete P-reducible metric, i.e. one which is not C-reducible. In order to do this, we study a class of Finsler metrics, called generalized P-reducible metrics, which contains the class of P-reducible metrics. We prove that every generalized P-reducible (α,β)-metric with vanishing S-curvature reduces to a Berwald metric or a C-reducible metric. It follows that there is no concrete P-reducible (α,β)-metric with vanishing S-curvature.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap114-1-5,
author = {A. Tayebi and H. Sadeghi},
title = {Generalized P-reducible ($\alpha$,$\beta$)-metrics with vanishing S-curvature},
journal = {Annales Polonici Mathematici},
volume = {113},
year = {2015},
pages = {67-79},
zbl = {1327.53100},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap114-1-5}
}
A. Tayebi; H. Sadeghi. Generalized P-reducible (α,β)-metrics with vanishing S-curvature. Annales Polonici Mathematici, Tome 113 (2015) pp. 67-79. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap114-1-5/