Journal of Mathematics of Kyoto University

A global approach to the theory of special finsler manifolds

Nabil Labib Youssef, Sayed Hassan Abed, and Amr Soleiman

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The aim of the present paper is to provide a \emph{global} presentation of the theory of special Finsler manifolds. We introduce and investigate\emph{ globally} (or intrinsically, free from local coordinates) many of the most important and most commonly used special Finsler manifolds\,: locally Minkowskian, Berwald, Landesberg, general Landesberg, $P$-reducible, $C$-reducible, semi-$C$-reducible, quasi-$C$-reducible, $P^{*}$-Finsler, $C^{h}$-recur-rent, $C^{v}$-recurrent, $C^{0}$-recurrent, $S^{v}$-recurrent, $S^{v}$-recurrent of the second order, $C_{2}$-like, $S_{3}$-like, $S_{4}$-like, $P_{2}$-like, $R_{3}$-like, $P$-symmetric, $h$-isotropic, of scalar curvature, of constant curvature, of $p$-scalar curvature, of $s$-$ps$-curvature.

The global definitions of these special Finsler manifolds are introduced. Various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced.

Although our investigation is entirely global, we provide; for comparison reasons, an appendix presenting a local counterpart of our global approach and the {\it{local}} definitions of the special Finsler spaces considered.

Article information

J. Math. Kyoto Univ., Volume 48, Number 4 (2008), 857-893.

First available in Project Euclid: 14 August 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C60: Finsler spaces and generalizations (areal metrics) [See also 58B20] 53B40: Finsler spaces and generalizations (areal metrics)


Youssef, Nabil Labib; Abed, Sayed Hassan; Soleiman, Amr. A global approach to the theory of special finsler manifolds. J. Math. Kyoto Univ. 48 (2008), no. 4, 857--893. doi:10.1215/kjm/1250271321.

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