On real and complex Berwald connections associated to strongly convex weakly Kähler–Finsler metric

In this paper, we give a definition of weakly complex Berwald metric and prove that, (i) a strongly convex weakly Kähler–Finsler metric F on a complex manifold M is a weakly complex Berwald metric iff F is a real Berwald metric; (ii) assume that a strongly convex weakly Kähler–Finsler metric F   is a weakly complex Berwald metric, then the associated real and complex Berwald connections coincide iff a suitable contraction of the curvature components of type (2,0)(2,0) of the complex Berwald connection vanish; (iii) the complex Wrona metric in CnCn is a fundamental example of weakly complex Berwald metric whose holomorphic curvature and Ricci scalar curvature vanish identically. Moreover, the real geodesic of the complex Wrona metric on the Euclidean sphere S2n−1⊂CnS2n−1⊂Cn is explicitly obtained.