Fibrewise Real-Analytic Landsberg Metrics in Dimension at Least Four Are Berwald

Introduction and main results

A Finsler metric replaces the inner-product norm of a Riemannian metric by a strongly convex Minkowski norm on each tangent space. Its canonical spray coefficients need not be quadratic in the fibre variables. A Berwald metric has spray coefficients quadratic in those variables. A Landsberg metric has vanishing Landsberg curvature; equivalently, its Berwald tensor is completely symmetric after its upper index is lowered with the fundamental tensor. Every Berwald metric is Landsberg. The converse is the Landsberg–Berwald problem, also called the unicorn problem. All metrics considered here are smooth and strongly convex on the entire slit tangent bundle. This regularity in all nonzero directions is an essential part of the problem. For the basic definitions we refer to [3]; for the history, principal methods and related developments, see the surveys of Feng–Li [12] and Crampin [8].

Matsumoto’s 1996 survey collected classical reduction results and asked for explicit examples of Landsberg metrics which are not Berwald [26 p. 82]. Bao called such metrics “unicorns” in his 2007 survey [2]. The existence question also appears as Problem 1 in Shen’s 2009 list of open problems [34].

Rigidity results for regular Landsberg metrics have been obtained under additional curvature assumptions and within special classes of metrics. A classical result states that a Landsberg metric of dimension $n \ge3$ with vanishing Douglas curvature is Berwald [26 Theorem 7]. Numata proved that a Landsberg space of dimension $n \ge3$ with nowhere-vanishing scalar flag curvature is a Riemannian space of constant sectional curvature [28]. In his systematic study of $(\alpha,\beta)$-metrics, Shen proved that every regular Landsberg $(\alpha,\beta)$-metric in dimension $n \ge3$ is Berwald [33 Theorem 1.1]. Starting from equivalence properties of the unit indicatrices of Minkowski norms, Feng–Han–Li extended this conclusion to general $(\alpha,\beta)$-metrics in dimension $n \ge4$ [11]. Xu–Deng proved that every Landsberg metric of $(\alpha_1,\alpha_2)$-type is Berwald, using a linear Landsberg equation associated with a fixed Minkowski norm and the action of its linear isometry group on the solution space [39 Theorem 1.2 and Section 5].

Another approach concerns the relations of the Landsberg condition to mean curvatures and volume. Using centroaffine geometry, Li established equivalence theorems for Minkowski spaces and proved that a Landsberg metric with vanishing mean Berwald curvature is Berwald in dimension $n \ge3$ [20, 19]. Crampin showed that it is enough for the trace of the mean Berwald curvature with respect to the fundamental tensor to vanish [7].

Li–Zhang proved, for general Finsler metrics, that vanishing Berwald scalar curvature implies vanishing mean Berwald curvature, giving another derivation of this Berwald criterion for Landsberg metrics [22 Theorems 1 and 2]. In 2024 Crampin further proved that, on a compact manifold without boundary of dimension $n \ge3$, a regular Landsberg metric whose Berwald scalar curvature is invariant under the geodesic flow is Berwald [8 Theorem 4]. The noncompact version also requires forward or backward geodesic completeness and boundedness of the scalar $g^{ij}\partial_{y^i}\partial_{y^j}\log\det(g_{ab})$ on the unit indicatrix bundle. In 2025, using integral identities on that bundle, Li proved that a Landsberg metric on a closed connected manifold is Berwald if its Berwald scalar curvature does not change sign on the entire unit indicatrix bundle [21 Theorem 3.7].

Nonregular examples and generalized structures exhibit further features of the Landsberg equations. Shen also constructs almost regular Landsberg $(\alpha,\beta)$-metrics which are not Berwald; his explicit example is singular in two extremal directions in each tangent space [33 Theorem 1.2 and Example 1.1]. Further constructions and related results are surveyed in [37]. By Cartan–Kähler theory and path geometry, Sabau–Shibuya–Shimada established the existence of nontrivial generalized Landsberg structures [30]. Realizing such a structure as a regular Finsler metric on the entire slit tangent bundle remains a separate problem. A preprint submitted by Elgendi in October 2026 further studies nonregular, including almost regular, non-Berwald Landsberg metrics of general $(\alpha,\beta)$-type [9]. The domains or regularity of these constructions differ from the hypotheses of the present paper; they therefore do not furnish counterexamples to the regular unicorn problem. Szabó announced a proof in the general regular case in 2008, but his subsequent correction stated that the proof was incomplete. The conclusion retained there concerns the stronger dual Landsberg condition [35, 36].

Recent work also approaches the problem through sprays and curvature distributions. Elgendi–Muzsnay studied the image, kernel and rank of the Berwald curvature of Landsberg metrics, together with their relations to the holonomy distribution and metrizability of the geodesic spray, and obtained corresponding structural restrictions [10]. A 2026 preprint of Mao–Zhou gives a proof that every smooth, strongly convex, regular Landsberg metric on a closed three-dimensional manifold is Berwald, without assuming reversibility or real analyticity [25 Theorem 1.1]. Closedness enters the final compactness argument in that work, while dimension three is also used in the preceding analysis of the indicatrix and curvature algebra.

The unicorn problem is closely related to Hessian rigidity on a single Minkowski space. Laugwitz’s early work connected the geometry of Minkowski indicatrices with the theory of centroaffine hypersurfaces [16]. His subsequent conjecture asserts that, for a smooth, strongly convex Minkowski norm $F$ regular in all nonzero directions in dimension $N \ge3$, flatness of $g = \operatorname{Hess}(F^2/2)$ on the entire slit vector space implies that $F^2$ is a positive-definite quadratic form [17, 40]. Only positive homogeneity is assumed, so that $F(-y) \ne F(y)$ is allowed. By the cone decomposition of the Hessian metric, this is equivalent to ellipsoid rigidity when the centroaffine metric of the indicatrix has constant sectional curvature 1. The dimension assumption cannot be omitted: the Hessian metric of every two-dimensional Minkowski norm is locally flat, and the same conclusion is false in dimension two [40 Section 1.1].

Among the classical results on the Laugwitz conjecture, Brickell established rigidity in the absolutely homogeneous, or reversible, case [4]. By affine-geometric methods, Schneider extended the conclusion to the case in which the centroid of the unit ball is the origin [31]; see [12, 23] for related discussions and new proofs. These assumptions concern the symmetry of the norm or the volume centroid of its unit ball and need not hold for a general nonreversible norm. Xu–Matveev proved Laugwitz rigidity for smooth, strongly convex Minkowski norms with the standard block-diagonal $SO(k) \times SO(N-k)$ symmetry. Xu subsequently extended flat-Hessian rigidity, in dimension $N \ge3$, to Minkowski norms induced by isoparametric foliations of the standard sphere [40, 38]. Li–Zuo also proved that a smooth, strongly convex, closed indicatrix of dimension at least 2 is an ellipsoid centred at the origin if it has constant sectional curvature and its traceless Tchebychev operator is parallel with respect to the Levi-Civita connection of the centroaffine metric [23]. These results use symmetry, an isoparametric structure or an additional condition on a centroaffine tensor.

Mao–Zhou proved the Laugwitz conjecture in ambient dimension $n = 3$, or indicatrix dimension $m = 2$, for smooth strongly convex Minkowski norms regular on the entire slit space, without reversibility or real analyticity [25 Corollary 3.3]. Thus the three-dimensional case of Theorem A below is already covered by their result. Our proof treats all dimensions $n \ge3$ under the real-analyticity hypothesis.

The distinction between Hessian isometry and linear equivalence is central to the relation between these two problems. Li’s equivalence theorem states that, in dimension at least 3, a norm-preserving, positively homogeneous diffeomorphism of degree one is linear if it preserves both the Hessian metric and the Cartan trace form [20, 19]. Preservation of the Cartan trace form cannot simply be omitted. Xu–Matveev’s analysis of Hessian isometries between symmetric norms further shows the following: fix $1 \le k \le n-1$ and suppose that, on a Finsler manifold of dimension $n \ge3$, every tangent-space norm has the standard $SO(k) \times SO(n-k)$ symmetry in suitable linear coordinates. Then the Landsberg condition implies the Berwald condition [40 Corollary 1.10]. This smooth theorem includes several familiar special classes of metrics, but its fibrewise symmetry remains a substantive hypothesis.

We prove Berwald rigidity under fibrewise real analyticity in dimension $n \ge4$, without symmetry assumptions. Part I establishes two independent rigidity theorems for a single Minkowski space; Part II applies them to the angular Berwald quartic form and concludes by an argument on an open subset of the base manifold. We quote classical results and prove the lemmas specific to this argument.

The main theorem and its hypotheses

We say that $F$ is regular on the entire slit tangent bundle if it is smooth and strictly positive on all of $TM \setminus\{0\}$, satisfies $F(x,\lambda y)=\lambda F(x,y)$ for $\lambda>0$, and has positive-definite fundamental tensor

We denote the canonical spray coefficients by $G^a$ and the Berwald curvature by $B^a{}_{ijk}=\partial_{y^i}\partial_{y^j}\partial_{y^k}G^a$. Our main result is the following.

Analyticity is required only in the fibre variables at each fixed base point. The base manifold need not carry an analytic structure, and no uniform radius of analyticity over the fibres is assumed. We require neither compactness nor geodesic completeness of $M$, nor the identity $F(x,-y)=F(x,y)$. The compact space used in the proof is the unit indicatrix

Regularity in all nonzero directions and strong convexity imply that it is compact and diffeomorphic to $S^m$.

The conclusion of the main theorem allows non-Riemannian Berwald metrics. For example, a nonquadratic Minkowski norm independent of the base point defines a locally Minkowski metric whose canonical spray and Berwald curvature both vanish. Quadratic fibres arise in the proof only under the temporary assumption that the Berwald quartic form is nonzero on a fibre; they are an intermediate step in excluding that assumption.

Two analytic rigidity theorems for Minkowski spaces

Both rigidity theorems are stated on a single vector space. Let $V$ be a real vector space of dimension $n \ge3$, let $D$ be its standard flat affine connection, and let $F$ be a Minkowski norm on $V$. Put

We denote the Levi-Civita connection of $h$ by $\nabla^h$ and use

to define the self-adjoint contraction operator $C_X$. The cone structure of the Hessian metric and the Gauss equation of the indicatrix take the form

where $\rho=F$ and $R_0(X,Y)Z=h(Y,Z)X-h(X,Z)Y$. In particular, $g$ is flat if and only if $h$ has constant sectional curvature $1$.

The first result is analytic Laugwitz rigidity, which we shall also call Theorem A.

Theorem A is restated and proved as Theorem 4.11. Flatness reduces the problem to a real-analytic cubic form on the entire standard sphere $(S^m,h_0)$ satisfying

where $m=n-1$ and $\nabla^{h_0}$ is the Levi-Civita connection of $h_0$. An extremum of the trace potential gives a point where the cubic tensor vanishes. Near this point, separation of the leading term of a homogeneous potential and the use of nested normal ideals give the local zero-subsphere theorem, Theorem 3.18. The first nonzero orders in different transverse directions may differ. The full collection of Gram relations bounds the total number of active axes by $2R\le m+1$, so that the induction terminates after finitely many steps. We then fix an ambient normal along a great-circle arc on which the cubic tensor vanishes and obtain a differential equation for an analytic periodic scalar function. Integrating this equation over one period gives a sum of nonnegative integrals equal to zero. The scalar function must therefore vanish, which proves global rigidity of the cubic form.

The second result deduces Hessian flatness from a twisted Codazzi equation on the indicatrix. We shall also call it Theorem B.

Theorem B requires only smoothness of $B$. It need not be real-analytic or arise from the Berwald curvature of a Finsler metric. Equation (1) and complete symmetry already imply the algebraic compatibility identity used below; that identity is not an additional hypothesis. The analyticity assumption concerns $F$ and the tensors $h$, $C$ determined by it. Reversibility of $F$ is not required. Theorem B is proved in Section 8 without using Theorem A.

To describe the two cases in the proof of Theorem B, put $m=n-1$ and denote by

the maximal rank of $B$. Since $B\not\equiv0$, we have $1\le q\le m$, and the points of maximal rank form a nonempty open set. If $0<q<m$, the radical distribution, the centroid of the restricted quartic form, Riccati transport along centroaffine geodesics, and the restrictions imposed by closed orbits imply that $C$ vanishes on a nonempty open set. Since $F$ is real-analytic, the vanishing of $C$ extends over the connected indicatrix. The Gauss equation and cone structure then give flatness of $g$. This branch proves the conclusion of Theorem B directly; it is unnecessary first to exclude a nonzero quartic form of deficient maximal rank.

If $q=m$, we first establish generic irreducibility of the contraction operators on the set where the radical is trivial. On a fixed positive-definite inner-product space, the angular algebraic compatibility identity is converted, by a full contraction identity, into curvature compatibility of each $B_{uv}$ with $K=[C,C]$. The theorem of Mantiča–Molinari on curvature-compatible tensors gives Jordan closure [24 arXiv version, Theorem 4.2]; we quote this result, specifying the index and curvature conventions needed here. An extremum of a nonzero quartic form on the unit sphere provides a contraction operator with a simple extreme eigenvalue, and hence a rank-one spectral projection. Irreducibility and this projection imply that the generated Jordan algebra consists of all self-adjoint operators. It follows that

(Theorem 7.17). The Gauss equation gives $R^{h}=\kappa R_{0}$, where $\kappa=1-s$. The real-analytic identity theorem extends this curvature isotropy from any nonempty open set to the entire indicatrix. Since $m\ge3$, Schur’s lemma makes $\kappa$ constant; an extremum of the trace potential and a deformation of the quartic then show that $\kappa=1$. Thus this branch also gives flatness of $g$, without using a low-dimensional classification of Jordan algebras.

In Part II we apply the two theorems for a single fibre to actual Landsberg metrics. The Berwald tensor with its upper index lowered,

is completely symmetric precisely when the Landsberg condition holds. Under this condition it vanishes whenever a radial vector is inserted, and its tangential restriction to each indicatrix satisfies (1.1). The set $\mathcal{U}=\{x\in M:B(x,\cdot)\ne0\}$ is open. At every $x\in\mathcal{U}$, the angular quartic form $B_x$ is nonzero, so Theorem B makes the fibre Hessian metric flat and Theorem A makes $F_x^2$ quadratic. Smoothness of the fundamental tensor then gives a Riemannian metric on $\mathcal{U}$. Its Berwald curvature must vanish, contradicting the definition of $\mathcal{U}$ unless this set is empty. This proves the main theorem.

The planarity and completeness of centroaffine geodesics are taken, respectively, from [29 Proposition 5.1 and Theorem 5.4]. Section 2.6 supplies the explicit parameter, periodicity of position and velocity, and uniform bounds needed for the subsequent transport argument.

We use the Einstein summation convention: repeated upper and lower indices in a term are summed over $1,\ldots,n$.

Scope and organization of the paper