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On Randers metrics of quadratic Riemann curvature. (English) Zbl 1171.53020

Geodesics of a Finsler space \(F^n\) with Finsler metric \(\mathcal F(x,y)\) have the equation
\[ \frac{d^2 x^i}{dt^2}+2G^i(x,y)=0,\qquad y=\frac{dx}{dt}. \]
If \(F^n\) is a Riemann space, then \(G^i=\Gamma_j{}^i{}_k(x)y^iy^j\), and \(G^i\) is quadratic in \(y\). Consequently, also \(R^i_k(x,y)\) is quadratic in \(y\). Finsler metrics with this property are called \(R\)-quadratic metrics. Berwald metrics are \(R\)-quadratic, but not only they. The paper presents a clear survey of \(R\)-quadratic Finsler metrics. Similarly are defined Ricci-quadratic and \(W\)-quadratic Finsler metrics (\(W\) relates to the projective Weyl curvature tensor.) In this paper, Randers metrics \(\mathcal F=\alpha+\beta\), \(\alpha=\sqrt{a_{ik}(x)y^iy^j}\), \(\beta=b_i(x)y^i\) with \(R\)-quadratic Riemann curvature are characterized by two equations involving \(\alpha\), \(\beta\), and \(b_{i|k}\). It is shown that \(R\)-quadratic Randers metrics have constant \(S\)-curvature. Also Randers metrics with Ricci-quadratic or \(W\)-quadratic curvature are characterized. Also several further theorems, corollaries and examples are given.

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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References:

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