On Berwald Spaces with non-Zero Flag Curvature

Abstract: We prove that every Berwald manifold with non-zero flag curvature is Riemannian. This result provides an extension of Numata and Szabo's rigidity theorems. We show that every positively curved constant isotropic Berwald manifold is Riemannian or locally Minkowskian. Then, we prove that every compact strictly positive (or negative) isotropic Berwald manifold reduces to a Berwald manifold. Finally, we prove that every homogeneous isotropic Berwald metric is either locally Minkowskian, or Riemannian, or Berwald metric or Berwald-Randers metric generalizing result previously only known in the case of Randers metric.

• The paper demonstrates that every Berwald manifold with vanishing flag curvature is locally Minkowskian, and those with nonzero flag curvature are Riemannian, removing classical dimension and scalar curvature assumptions.
• Constant isotropic Berwald metrics with positive curvature are proven to be Riemannian, while those with negative curvature can be non-Riemannian.
• Compact constant isotropic Berwald manifolds are shown to be Berwald, and homogeneous isotropic Berwald metrics are classified.

Overview

This paper, by A. Tayebi and B. Najafi (2601.20087), establishes a family of rigidity results for Finsler manifolds whose Berwald curvature is either zero or isotropic. The central theorem removes dimension and scalar-curvature hypotheses from classical rigidity theorems of Numata and Szabó: every Berwald manifold with flag curvature vanishing for all flags is locally Minkowskian, and every Berwald manifold with nowhere-zero flag curvature is Riemannian. The paper then extends this line to constant isotropic Berwald metrics, compact strictly positive (or negative) isotropic Berwald metrics, and homogeneous isotropic Berwald metrics.

Background and motivation

Finsler geometry carries both Riemannian curvatures (Riemann curvature, flag curvature) and non-Riemannian ones (Cartan torsion, Berwald curvature, Landsberg curvature, $S$-curvature). The Berwald curvature $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$ measures the dependence of spray coefficients on fiber variables; ${\bf B}=0$ characterizes Berwald metrics, which are affinely equivalent to Riemannian metrics and for which all tangent Minkowski spaces $(T_xM, F_x)$ are linearly isometric along parallel translation (Ichijyō).

Numata's 1975 theorem states that an $n$-dimensional ($n\geq 3$) Berwald manifold of scalar flag curvature is locally Minkowskian if ${\bf K}$ vanishes on $TM_0$ and Riemannian if ${\bf K}$ is nowhere zero; Szabó's 1981 result completes this in dimension two, showing every Berwald surface is either locally Minkowskian or Riemannian. Both results require the scalar flag curvature hypothesis. The main contribution here is its removal.

Rigidity of Berwald manifolds without scalar curvature assumption

The key identity underlying the argument relates the mean Landsberg curvature $\bf J$, mean Cartan torsion $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$0, $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$1-curvature, and Riemann curvature:

$${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$2

For a Berwald metric, $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$3 and $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$4, so the identity collapses to $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$5. Since $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$6 is $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$7-orthogonal to $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$8, any nonzero $${\bf B}^i{}_{jkl} = \partial^3 G^i/\partial y^j \partial y^k \partial y^l$$9 spans with ${\bf B}=0$0 a flag ${\bf B}=0$1 whose flag curvature must vanish — contradicting the nowhere-zero hypothesis (flag curvature is continuous, hence has fixed sign). Deicke's theorem then forces ${\bf B}=0$2, i.e., ${\bf B}=0$3 is Riemannian. If instead ${\bf B}=0$4 vanishes identically, standard arguments give local Minkowski structure. This yields the main rigidity theorem without any dimension restriction or scalar-curvature assumption, and as a corollary gives a short new proof of Szabó's surface theorem via Akbar-Zadeh's result that Berwald surfaces have pointwise-constant flag curvature.

A companion corollary replaces the Berwald condition: a weakly Landsberg manifold (${\bf B}=0$5) with constant mean Berwald curvature satisfies the same dichotomy. This partially answers whether Wu's closedness condition for weakly Landsberg manifolds of negative flag curvature can be traded for other curvature hypotheses.

Positively curved constant isotropic Berwald metrics

Motivated by Funk metrics (solutions of Hilbert's fourth problem satisfying ${\bf B}=0$6), Cheng–Shen introduced isotropic Berwald metrics, where ${\bf B}=0$7 takes the form involving the angular form ${\bf B}=0$8, Cartan torsion ${\bf B}=0$9, and a function $(T_xM, F_x)$0; constant isotropic means $(T_xM, F_x)$1 is constant. Berwald metrics correspond to $(T_xM, F_x)$2 and Funk metrics to $(T_xM, F_x)$3.

The authors prove that a positively curved non-zero constant isotropic Berwald manifold is Riemannian. The proof exploits that such metrics have $(T_xM, F_x)$4, $(T_xM, F_x)$5, $(T_xM, F_x)$6, and satisfy $(T_xM, F_x)$7; contracting with $(T_xM, F_x)$8 forces $(T_xM, F_x)$9 wherever $n$0, contradicting positive curvature. A corollary covers positively curved Douglas metrics with non-zero constant isotropic mean Berwald curvature.

Notably, the authors state plainly that this result fails for negative flag curvature: the standard Funk metric on $n$1 is a non-Riemannian Randers-type constant isotropic Berwald metric with $n$2. So positivity is essential, not incidental.

Compactness forces triviality

For a compact manifold with strictly positive (or negative) isotropic Berwald curvature ($n$3), the distortion $n$4 associated with the Busemann–Hausdorff volume form is bounded on the compact unit sphere bundle, yet evolves along unit-speed geodesics as $n$5, forcing linear growth incompatible with boundedness. Hence $n$6 and $n$7 is a Berwald metric. Consequently, every compact constant isotropic Berwald manifold is Berwald. This is a clean global rigidity statement obtained from volume-form distortion alone, independent of flag curvature assumptions.

Homogeneous isotropic Berwald metrics

Using Deng–Hou's theorem that the isometry group of a Finsler manifold is a Lie transformation group, the authors classify homogeneous isotropic Berwald metrics on connected manifolds: each is either locally Minkowskian, Riemannian, Berwald, or a Randers metric of Berwald type. In dimension two this follows because homogeneous Finsler metrics with isotropic $n$8-curvature have vanishing $n$9-curvature (Xu–Deng), forcing $n\geq 3$0 and reducing to Szabó's classification; in dimensions $n\geq 3$1 it follows from the authors' earlier work. The theorem thus generalizes Deng–Hu's 2013 result for homogeneous Randers spaces of Berwald type, which had been conjectured (but unproven) in dimension two in prior work of the authors.

Limitations and open questions

Several restrictions are intrinsic to the results rather than artifacts of technique. The positivity hypothesis in the constant isotropic case cannot be dropped, as the Funk metric example demonstrates; whether an analogous classification holds for negatively curved constant isotropic Berwald manifolds beyond the Randers/Funk examples remains open. The compactness theorem requires strict sign of $n\geq 3$2; the behavior when $n\geq 3$3 changes sign or vanishes on part of $n\geq 3$4 is not addressed. The weakly Landsberg corollary requires constant mean Berwald curvature, leaving open whether closeness in Wu's theorem can be replaced by weaker non-Riemannian curvature conditions. Finally, the homogeneous classification relies on existing structural results in dimensions $n\geq 3$5 and on Szabó's theorem in dimension two, so it inherits whatever gaps those classifications contain.

Conclusion

The paper consolidates and extends the rigidity theory of Berwald-type Finsler metrics. Its principal achievement is the dimension-free, scalar-curvature-free dichotomy for Berwald manifolds (locally Minkowskian versus Riemannian), derived from a single identity linking $n\geq 3$6, $n\geq 3$7, $n\geq 3$8, and $n\geq 3$9. The subsequent results show that positivity plus isotropy of Berwald curvature forces Riemannian structure, compactness plus strict isotropy forces Berwald structure, and homogeneity admits only the four expected model classes. Together these sharpen the boundary between genuinely Finslerian behavior and Riemannian rigidity in a precise, checkable way.