Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 32, No. 1, pp. 125-133 (2016)

Ricci Tensor of a Finsler space with special $(\alpha,\beta)$-metrics

Roopa Mallappa Kariyappa and Narasimhamurthy Senajji Kampalappa

Kuvempu University

Abstract: In this paper, we investigate the Ricci tensor of a Finsler space of a special $(\alpha, \beta)$-metric $F=\frac{(\alpha+\beta)^{2}}{\alpha}+\beta$, where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ be a Riemannian metric and $\beta$ be a 1-form. We also prove that if $\alpha$ is a positive (negative) sectional curvature and $F$ is of $\alpha$-parallel Ricci curvature with constant Killing 1-form $\beta$, then $(M, F)$ is a Riemannian Einstein space.

Keywords: Finsler space, $(\alpha, \beta)$-metrics, Ricci tensor, Einstein space, 1-form, Ricci curvature

Classification (MSC2000): 53B40; 53C20, 53C60

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