De Rham Decomposition Theorem for Strongly Convex Kähler–Berwald Manifolds

In this paper, we prove that the unit ball Bn⊂Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_n\subset \mathbb {C}^n$$\end{document} admits no Aut(Bn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Aut }(B_n)$$\end{document}-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the canonical Poincaré–Bergman metric, while the unit polydisk Pn⊂Cn(n≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_n\subset \mathbb {C}^n(n\ge 2)$$\end{document} admits infinite many Aut(Pn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Aut }(P_n)$$\end{document}-invariant strongly convex complex Finsler metrics other than the Bergman metric. The Aut(Pn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Aut }(P_n)$$\end{document}-invariant strongly pseudoconvex complex Finsler metrics (which are not necessary Hermitian quadratic) are explicitly constructed and are proved to be strongly convex Kähler–Berwald metrics. We also investigate the existence of holomorphic invariant strongly pseudoconvex non-Hermitian quadratic complex Finsler metrics on reducible complex manifolds, and give a de Rham decomposition theorem for strongly convex Kähler–Berwald manifolds.