Complex Berwald Metric Overview

• Complex Berwald metrics are strongly pseudoconvex complex Finsler metrics characterized by constant Christoffel symbols that ensure flat holomorphic sectional curvature.
• On complex Lie groups, left-invariant metrics naturally become Berwald metrics, exhibiting rigidity in structure and curvature due to the algebraic properties of the Lie algebra.
• In the abelian case, these metrics satisfy Kähler conditions, and U(n)-invariant models further reveal unique geodesic behavior and curvature vanishing results.

A complex Berwald metric is a distinguished class of strongly pseudoconvex complex Finsler metrics, characterized by rigid parallel translation and curvature properties analogous to the Berwald condition in real Finsler geometry. Within the context of complex manifolds and particularly on complex Lie groups, these metrics exhibit a pronounced structural rigidity, unifying the theory with explicit criteria for Kähler and curvature properties, and forming a central theme in contemporary research on complex Finsler geometry (Xu et al., 22 Dec 2025, Luo et al., 31 Dec 2025).

1. Formal Definition and Characterization

Let $M$ be a complex manifold of complex dimension $n$ with holomorphic tangent bundle $T^{1,0}M$. A strongly pseudoconvex complex Finsler metric is a continuous function

$F: T^{1,0}M \to [0,\infty)$

which restricts at each $z \in M$ to a complex Minkowski norm on $T^{1,0}_z M$: $F_z(\lambda v) = |\lambda| F_z(v)$ for $\lambda \in \mathbb{C}$, $F_z$ smooth away from $0$, and its Levi matrix $$G_{i\bar{j}} = \frac{\partial^2}{\partial w^i \partial \bar{w}^j}(F^2)$$ is positive definite for all $w \neq 0$. The Chern-Finsler connection is constructed via the fundamental tensor $G_{i\bar{j}}$, with connection (Christoffel) symbols

$$\Gamma^{i}_{jk} = \frac{\partial N^i_k}{\partial w^j}$$

where $N^i_k$ are the nonlinear connection coefficients built from $G_{i\bar{j}}$.

A Finsler metric $F$ is called a complex Berwald metric if $\Gamma^{i}_{jk}$ are independent of the fiber variable $w$ (or, equivalently, the $v$-variable in left-invariant settings). This equivalently requires the vanishing of the $h$-curvature of the Chern-Finsler connection or the holomorphic extension of the canonical complex spray (Xu et al., 22 Dec 2025, Luo et al., 31 Dec 2025).

2. Left-Invariant Complex Berwald Metrics on Lie Groups

Let $G$ be a complex Lie group of dimension $n$, with complex Lie algebra $\mathfrak{g}^{1,0}$. Any left-invariant strongly pseudoconvex complex Finsler metric $F$ on $G$ is determined by its restriction to $T^{1,0}_e G \cong \mathfrak{g}^{1,0}$, and all geometric data (fundamental tensor, connection, torsion) are functions of the fiber variable $v \in \mathfrak{g}^{1,0} \setminus \{0\}$ only (Luo et al., 31 Dec 2025).

For left-invariant $F$, the Christoffel symbols $\Gamma^{i}_{jk}$ are constant in $v$, due to the algebraic properties of complex Lie groups: specifically, $[\mathfrak{g}^{1,0}, \mathfrak{g}^{0,1}] = 0$ implies that the connection coefficients depend only on the structure constants of the Lie algebra and not on $v$. Consequently, all left-invariant complex Finsler metrics on $G$ are complex Berwald metrics (Xu et al., 22 Dec 2025, Luo et al., 31 Dec 2025).

3. Spray Structure, Realification, and Holomorphic Extension

The canonical complex spray associated to a complex Finsler metric $F$ is given by

$\chi = w^{k} \delta_{z^{k}}$

where $$\delta_{z^{k}} = \partial_{z^{k}} - N^{i}_{k} \partial_{w^{i}}$$. In the left-invariant setting, this spray extends holomorphically to all of $T^{1,0}G$, and in terms of right-invariant frames, is simply $\chi = v^{i} \tilde{V}_{i}$.

The realification map $(\cdot)^{\circ} : T^{1,0} G \to TG$ takes the real part of $\chi$ to the canonical bi-invariant spray (i.e., the unique left- and right-invariant spray on the real Lie group $G$). Explicitly, for $v^i = v^i_\mathbb{R} + i v^i_\mathbb{I}$,

$$\chi^{\circ} = v^i_\mathbb{R} \tilde{V}_i^{\circ} + v^i_\mathbb{I} J(\tilde{V}_i^{\circ}).$$

This identification links the holomorphic geometry of $T^{1,0} G$ directly to classical real Lie group geometry (Xu et al., 22 Dec 2025).

4. Curvature Properties and Rigidity Results

For a general complex Finsler manifold, the holomorphic sectional curvature along $w$ is derived from the curvature tensor

$$R_{i\bar{j} k\bar{\ell}} = -\frac{\partial^2}{\partial z^k \partial \bar{z}^{\ell}} G_{i\bar{j}} + G^{\bar{q} p} \left( \frac{\partial G_{i\bar{q}}}{\partial z^{k}} \frac{\partial G_{p\bar{j}}}{\partial \bar{z}^{\ell}} \right).$$

For any left-invariant complex Berwald metric $F$ on a complex Lie group $G$, a direct calculation shows that for all $w$,

$$w^i \bar{w}^j w^k \bar{w}^\ell R_{i\bar{j} k\bar{\ell}} = 0$$

and thus the holomorphic sectional curvature $K(w)$ vanishes identically. Similarly, the bisectional curvature $B(v, w) = 0$ for all $v, w \in \mathfrak{g}^{1,0}$ (Xu et al., 22 Dec 2025, Luo et al., 31 Dec 2025).

This vanishing is a rigidity phenomenon unique to the complex Lie group setting: all left-invariant strongly pseudoconvex complex Finsler metrics are complex Berwald metrics with flat holomorphic sectional and bisectional curvature (Luo et al., 31 Dec 2025).

5. Kähler and Weakly Kähler Conditions; Abelian Criterion

Complex Finsler geometry distinguishes three levels of Kähler condition:

• Strongly Kähler: $\Gamma^{i}_{j;k} - \Gamma^{i}_{k;j} = 0$ everywhere,
• Kähler: $w^k (\Gamma^{i}_{j;k} - \Gamma^{i}_{k;j}) = 0$,
• Weakly Kähler: $w^k (\Gamma^{i}_{j;k} - \Gamma^{i}_{k;j})_i= 0$.

In the left-invariant context, these three notions are equivalent: they hold if and only if the Lie algebra $\mathfrak{g}^{1,0}$ is abelian, i.e., all structure constants vanish. In this case, $F$ is said to be a Kähler-Berwald metric and $G$ is holomorphically isomorphic to $(\mathbb{C}^n, +)$ or, up to covers, to a complex torus (Xu et al., 22 Dec 2025, Luo et al., 31 Dec 2025).

On non-abelian complex Lie groups, left-invariant complex Finsler metrics are Berwald but never Kähler. The vanishing of torsion (the necessary and sufficient condition for Kähler-Berwald) is algebraically equivalent to the vanishing of the Lie bracket.

6. U(n)-Invariant Complex Berwald Metrics and Related Phenomena

On domains in $\mathbb{C}^n$ with $U(n)$ symmetry, complex Berwald metrics appear as a rigid subclass of all $U(n)$-invariant complex Finsler metrics. The main results are:

• $U(n)$-invariant real Berwald complex Finsler metrics coincide with those arising from Hermitian quadratic forms. Any non-Hermitian $U(n)$-invariant complex Finsler metric cannot be Berwald (Wang et al., 2020).
• The characterization of weakly complex Berwald metrics with vanishing holomorphic sectional curvature is explicit: $F$ is such if and only if the underlying function $\varphi(t,s)$ satisfies $\varphi=f(s-t)$ for some smooth positive $f$, and in these cases both complex spray coefficients are quadratic and the curvature vanishes identically.
• All real geodesics of $U(n)$-invariant weakly complex Berwald metrics, when restricted to the unit sphere $\mathbb{S}^{2n-1}\subset \mathbb{C}^n$, are great circles with the same length as in the standard Hermitian metric, reflecting further rigidity (Wang et al., 2020).

7. Summary Table: Structural Properties of Left-Invariant Complex Berwald Metrics

Property	General Left-Invariant Complex Finsler	Abelian (Kähler-Berwald) Case
Connection coefficients $\Gamma^i_{jk}$	Constant in $v$ (fiber direction)	Constant, torsion-free
Holomorphic sectional curvature	$0$	$0$
Kähler property	Equivalence of strong, regular, weak	All satisfied iff Lie algebra abelian
Realification of spray	Bi-invariant spray on $G$	Bi-invariant spray

All entries are deduced from (Xu et al., 22 Dec 2025, Luo et al., 31 Dec 2025).

• Left invariant complex Finsler metrics on a complex Lie group (Xu et al., 22 Dec 2025)
• Curvature of left-invariant complex Finsler metric on Lie groups (Luo et al., 31 Dec 2025)
• On $U(n)$-invariant strongly convex complex Finsler metrics (Wang et al., 2020)