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 $(\alpha , \beta )$-metric with vanishing S-curvature reduces to a Berwald metric or a C-reducible metric. It follows that there is no concrete P-reducible $(\alpha ,\beta )$-metric with vanishing S-curvature.
Keywords:
study problems finsler geometry presented matsumoto shimada about existence concrete p reducible metric which c reducible order study class finsler metrics called generalized p reducible metrics which contains class p reducible metrics prove every generalized p reducible alpha beta metric vanishing s curvature reduces berwald metric c reducible metric follows there concrete p reducible alpha beta metric vanishing s curvature
Affiliations des auteurs :
A. Tayebi
1
;
H. Sadeghi
1
1
Department of Mathematics Faculty of Science University of Qom Qom, Iran
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author = {A. Tayebi and H. Sadeghi},
title = {Generalized {P-reducible} $(\alpha , \beta )$-metrics with vanishing {S-curvature}},
journal = {Annales Polonici Mathematici},
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year = {2015},
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A. Tayebi; H. Sadeghi. Generalized P-reducible $(\alpha , \beta )$-metrics with vanishing S-curvature. Annales Polonici Mathematici, Tome 114 (2015) no. 1, pp. 67-79. doi: 10.4064/ap114-1-5
