Let \(F: T^{1,0}M\rightarrow [0,+\infty )\) be a strongly convex complex Finsler metric on a complex manifold M, and let \(\pmb {J}\) denote the canonical complex structure on the complex manifold \(T^{1,0}M\) . In this paper, we provide a geometric characterization of strongly convex Kähler-Berwald metrics. Specifically, we prove that \(\pmb {J}\) is horizontally parallel with respect to the Cartan connection if and only if F is a Kähler-Berwald metric. Moreover, we show that the Cartan connection and the Chern-Finsler connection associated with F coincide if and only if \(\pmb {J}\) is both horizontally and vertically parallel with respect to the Cartan connection. Based on these results, we obtain a rigidity theorem for strongly convex Kähler-Berwald metrics with constant holomorphic sectional curvature.
