On unitary invariant weakly complex Berwald metrics with vanishing holomorphic curvature and closed geodesics

Abstract

In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form $F = \sqrt {rf\left( {s - t} \right)}$, where $r = {\left\| v \right\|^2}$, $s = \frac{{{{\left| {\left\langle {z,v} \right\rangle } \right|}^2}}}{r}$, $t = {\left\| z \right\|^2}$, f(w) is a real-valued smooth positive function of w ∈ R, and z is in a unitary invariant domain M ⊂ Cⁿ. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f′, we prove that all the real geodesics of $F = \sqrt {rf\left( {s - t} \right)}$ on every Euclidean sphere S²ⁿ⁻¹ ⊂ M are great circles.