Geodesic
On projective Ricci curvature of ‎Matsumoto‎ ‎metrics
Mehran Gabrani1 ; Bahman Rezaei1 ; Akbar Tayebi2 ; Mehran Gabrani ; Bahman Rezaei ; Akbar Tayebi
1Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran
2Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran
Acta mathematica Universitatis Comenianae, Tome 90 (2021) no. 1, pp. 111-126 
Mehran Gabrani; Bahman Rezaei; Akbar Tayebi; Mehran Gabrani; Bahman Rezaei; Akbar Tayebi. On projective Ricci curvature of ‎Matsumoto‎ ‎metrics. Acta mathematica Universitatis Comenianae, Tome 90 (2021) no. 1, pp. 111-126. http://geodesic.mathdoc.fr/item/AMUC_2021_90_1_a7/
@article{AMUC_2021_90_1_a7,
     author = {Mehran Gabrani and Bahman Rezaei and Akbar Tayebi and Mehran Gabrani and Bahman Rezaei and Akbar Tayebi},
     title = { On projective {Ricci} curvature of {‎Matsumoto‎} ‎metrics},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {111--126},
     year = {2021},
     volume = {90},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2021_90_1_a7/}
}
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Voir la notice de l'article provenant de la source Comenius University

In this paper, we study Finsler metrics with weak, isotropic or flat projective Ricci curvature (briefly, $\textbf{PRic}$-curvature). First, we prove a rigidity result that shows for a complete Finsler manifold inequality \ ${\bf PRic}\geq{\bf Ric}$ holds if and only if ${\bf S}=0$. Then, we show that the Matsumoto metric is of weak $\textbf{PRic}$-curvature if and only if it is a $\textbf{PRic}$-flat metric. We characterize projective Ricci flat Matsumoto metrics with constant length one-forms. In this case, we show that the Matsumoto metric reduces to a Ricci flat metric. Finally, we prove that a Matsumoto metric is $\textbf{PRic}$-reversible if and only if it is $\textbf{PRic}$-quadratic.