Projectively Equivalent Finsler Metrics

• Projectively equivalent Finsler metrics are defined as pairs sharing the same unparametrized geodesics, generalizing the Riemannian concept.
• Rigorous classification shows that in analytic settings, such as closed surfaces with negative Euler characteristic, these metrics differ by a constant factor and a closed 1-form.
• The study informs applications like Hilbert’s Fourth Problem and the construction of non-smooth metric families, highlighting the interplay between geometry, topology, and dynamics.

A projectively equivalent pair of Finsler metrics consists of two metrics whose unparametrized oriented geodesics coincide. In Finsler geometry, this notion generalizes the classical Riemannian concept, where projective equivalence implies that the metrics share the same geodesics up to orientation-preserving reparametrization. Rigorous classification results and structural theorems characterize projectively equivalent Finsler metrics in various settings, emphasizing analytic, topological, and dynamical constraints.

1. Definitions and Fundamental Concepts

A $C^\infty$ Finsler metric on a smooth manifold $M$ is a function %%%%2%%%% satisfying:

• Positive 1-homogeneity: $F(x, \lambda v) = \lambda F(x, v)$ for $\lambda > 0$.
• Smoothness on $TM \setminus 0$.
• Strong convexity: the fiber Hessian $$g_{ij}(x, v) = \frac{1}{2} \frac{\partial^2}{\partial v^i \partial v^j} F^2(x, v)$$ is positive definite for all $(x, v) \in TM \setminus 0$.

The geodesics of $F$ are the projections of integral curves of its geodesic spray $S$, defined locally as $$S = v^i \frac{\partial}{\partial x^i} - 2 G^i(x, v) \frac{\partial}{\partial v^i}$$, where the spray coefficients $G^i$ satisfy the Euler-Lagrange equations for $L = \frac{1}{2} F^2$.

Two Finsler metrics $F_1$ and $F_2$ are projectively equivalent if each geodesic (as an unparametrized oriented curve) of $F_1$ is also a geodesic of $F_2$. This equivalence is expressed via the relation between their sprays:

$G_2^i(x, v) = G_1^i(x, v) + P(x, v) v^i,$

for some scalar function $P(x, v)$, homogeneous of degree 1 in $v$ (Lang, 2019, Fukuoka, 2018).

2. Classification Results for Surfaces of Negative Euler Characteristic

On a closed, real-analytic surface $M$ with negative Euler characteristic, the classification of projectively equivalent real-analytic Finsler metrics is rigid: Theorem 1.1 (Lang, 2019): $F_1$ and $F_2$ are projectively equivalent if and only if there exist $c > 0$ and a closed $1$-form $\beta$ such that

$$F_2(x, v) = c F_1(x, v) + \beta_x(v) \qquad \forall (x, v) \in TM.$$

The proof utilizes:

• Proportionality of fiber Hessians: $h^{(1)}_{ij}(x, v) = c\, h^{(2)}_{ij}(x, v)$.
• The first integral $$I(x, v) = \frac{\mathrm{tr}\, h^{(1)}(x, v)}{\mathrm{tr}\, h^{(2)}(x, v)}$$ is constant along geodesics.
• Dynamical constraints: positive topological entropy for the geodesic flow (Dinaburg–Manning), and Paternain's entropy vanishing under real-analytic integrability, which force $I$ to be constant.

The closedness of $\beta$ emerges from the requirement that the difference $F_2 - c F_1$ yields geodesics compatible in the projective sense.

3. Projectively Equivalent Finsler Metrics Beyond Smooth and Analytic Categories

Projective equivalence extends to non-smooth metrics, notably $C^0$ Finsler structures. For instance, in (Fukuoka, 2018), an infinite-dimensional family of projectively equivalent $C^0$ Finsler metrics on $\mathbb{R}^2$ of the form $\hat F(x, y) = f(x) F_0(y)$ is constructed:

• $F_0$ is a norm with a regular hexagonal unit ball.
• The weight $f(x)$ is strictly positive, continuous, and monotonic in specified directions.

All metrics in this family share the same unparametrized geodesics, which are piecewise linear paths in distinguished directions. These spaces violate classical regularity properties such as Busemann convexity and fail to admit bounded strongly convex open sets.

4. Analytic and Topological Constraints for Projective Equivalence

Projective equivalence is analytically constrained by:

• The Rapcsák conditions, which require

$F_{x^i} - F_{v^i x^j} v^j + 2 G^j F_{v^i v^j} = 0,$

equivalent to sharing unparametrized geodesics (Lang, 2019).

Topologically, entropy arguments ensure rigidity in closed surfaces of negative Euler characteristic. Positive entropy for the geodesic flow excludes functionally independent real-analytic first integrals other than those forced by the structure, dictating proportionality of fiber Hessians and eventual form $F_2 = c F_1 + \beta$ with $\beta$ closed.

5. Projective Equivalence and Curvature: Flatness and Constant Flag Curvature

In the context of $(\alpha, \beta)$-metrics (metrics expressible as $$F(x, y) = \alpha(x, y) \phi(\beta(x, y)/\alpha(x, y))$$), projectively equivalent metrics are classified for both regular and singular cases. Classification in (Yang, 2013) shows:

• If both $\alpha$ and $\beta$ are flat-parallel, $F$ is locally Minkowskian and projectively flat.
• Certain singular metrics (Kropina, $m$-Kropina) are locally projectively flat with $K=0$, but need not be flat-parallel.
• Projectively flat metrics with constant flag curvature can be constructed explicitly, with necessary and sufficient conditions expressible via ODEs for $\phi(s)$.

In two dimensions, non-closedness of $\beta$ may occur for projectively flat $(\alpha, \beta)$-metrics, contrasting with higher-dimensional regularity results (Yang, 2013).

6. First Integrals and Invariant Structures in Projective Classes

Projectively equivalent Finsler metrics yield common first integrals of the geodesic flow. For two metrics $F$ and $\tilde F$, the characteristic polynomial of the endomorphism determined by their angular metrics produces $n-1$ nontrivial, fiberwise $0$-homogeneous first integrals, all invariant along geodesics of $F$ (Bucataru, 2021). In dimension $2$, this reduces to one essential invariant proportional to the ratio of angular metrics. These integrals are universal within the projective class.

7. Applications, Special Cases, and Geometric Significance

The rigidity result for analytic metrics on compact hyperbolic surfaces implies that every real-analytic projectively flat Finsler metric differs from any other by only dilation and addition of a closed $1$-form (Lang, 2019). Non-analytic examples on the sphere display greater flexibility, emphasizing the necessity of analyticity in the classification. Construction of large families of non-smooth projectively equivalent metrics (Fukuoka, 2018) provides new models for Hilbert's Fourth Problem and illustrates metric phenomena absent in the smooth category.

In summary, projectively equivalent Finsler metrics exhibit highly constrained forms under analytic and topological hypotheses, while non-smooth and non-analytic settings admit a richer variety of equivalence classes. The interplay between analytic integrability, dynamical entropy, and geometric structures governs the classification, rigidity, and flexibility within projective Finsler geometry.