A characterization of weakly Berwald spaces with \((\alpha,\beta)\)-metrics (Q2672310)

In Finsler geometry the concept of weakly Berwald metric \(F\) was introduced in [\textit{S. Bácsó} and \textit{R. Yoshikawa}, Publ. Math. 61, No. 1--2, 219--231 (2002; Zbl 1006.53022)] and means that \(F\) has vanishing \(\mathbf{E}\)-curvature. In the present paper the authors prove that on a smooth manifold \(M^n\), \(n\geq 3\), an \((\alpha, \beta)\)-metric \(F=\alpha\phi(s)\), \(s=\beta/\alpha\), where \(\alpha=\sqrt{a_{ij}(x)y^iy^j}\) is a Riemannian metric, \(\beta=b_i(x)y^i\) is a 1-form and the function \(\phi(s)\) is a real analytic but not even function, is a weakly Berwald metric if and only if it has vanishing \(\mathbf{S}\)-curvature.