Hermitian Finsler metrics and the Kobayashi metric
Given two Hermitian Finsler metrics, the author shows when we can obtain one from the other by a local bihomolomorphic change of coordinates. This problem is solved by using the method of equivalence due to E. Cartan [\textit{R. B. Gardner}, The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 58 (1989; Zbl 0694.53027)]. The second order invariants of smooth Hermitian Finsler metrics are interpreted here. The structure equations of the connection and the Euler-Lagrange equations of the length functional in terms of the connection are also given. The last result contains the converse of \textit{L. Lempert}'s theorem [Bull. Soc. Math. Fr. 109, 427-474 (1981; Zbl 0492.32025)] and it finds some differential conditions which imply that the Finsler metric is the Kobayashi metric of the underlying manifold.
scientific article; zbMATH DE number 4165923