Quadratic Curvature Properties in Metric Geometry

• Quadratic curvature properties are defined by quadratic forms derived from fibre Hessians and geodesic sprays, impacting geodesic equivalence and metric invariants.
• They enable classification of Finsler metrics via invariant integrals and projective changes, linking flag curvature and completeness with topological entropy.
• Applications include analyzing projectively flat (α,β)-metrics on spheres and ensuring global hyperbolicity in Lorentzian metrics through quadratic estimates.

Quadratic curvature properties in metric geometry pertain to aspects of curvature invariants, integrability, classification, and projective relations for Finsler and related metric structures, with a particular focus on quadratic forms (such as those arising from fibre Hessians and geodesic sprays). These properties are central in the analysis of geodesic equivalence, projective changes, completeness, flag curvature, and invariant integrals, with deep connections to entropy, topological invariants, and metric classification on manifolds of varying dimension and topology.

1. Projective Equivalence and Quadratic Structures

Projective (geodesic) equivalence for Finsler metrics is defined via the preservation of the family of oriented, unparametrized geodesics under metric deformation. For Finsler metrics $F, \tilde F : TM \to [0,\infty)$, projective equivalence holds if each oriented geodesic curve for $F$ is also a geodesic for $\tilde F$ after an orientation-preserving reparametrization.

For real-analytic Finsler metrics $\hat F, \check F$ on a closed surface $\mathcal S$ of negative Euler characteristic ($\chi(\mathcal S) < 0$), such equivalence is characterized by the existence of a scaling constant and addition of a closed 1-form: $$\check F(x,v) = \lambda\, \hat F(x,v) + \beta_x(v),$$ where $\lambda > 0$, and $\beta$ is closed. This results from the analysis of fibre Hessians (quadratic forms in tangent directions) and the Rapcsák-type geodesic spray equation, under which the traces of the Hessians are proportional and yield a first integral of the spray (Lang, 2019).

2. Quadratic Integrals and Invariant Volume Forms

Projectively equivalent Finsler metrics possess families of 0-homogeneous first integrals on the projective sphere bundle $SM$, arising as coefficients of a characteristic polynomial generated from the quadratic angular metrics and fundamental tensors. For $F, \bar F$ projectively equivalent on an $n$-manifold, the tensor

$$H^i{}_j = g^{ik}(\bar{h}_{kj} - (\bar F / F)^2 h_{kj})$$

has eigenvalues whose characteristic polynomial yields invariant functions $f_a(x,y)$, each 0-homogeneous and constant along geodesic flows. These are universal within the projective class. In two dimensions, the angular metric reduction yields explicit closed forms, aligning with classical Liouville-type integrals. Such invariants encode quadratic curvature properties intrinsically tied to the metric's projective geometry and integrability (Bucataru, 2021).

3. Entropy, Euler Characteristic, and Integrability

A closed surface with negative Euler characteristic ($\chi(\mathcal S) < 0$) features a fundamental group of exponential growth and thus induces positive topological entropy in the geodesic flow of any Finsler metric on $\mathcal S$. Conversely, for analytic Hamiltonian systems on $T\mathcal S \setminus 0$ with a nontrivial analytic first integral (not depending on energy), entropy is forced to vanish. Reconciling these facts imposes constancy of quadratic ratios (such as $$I(x,\xi) = \textrm{tr}\,\hat h / \textrm{tr}\,\check h$$), leading to proportionality of fibre Hessians and thus to the aforementioned classification—demonstrating the deep interplay between quadratic curvature invariants, topological entropy, and analytic integrability (Lang, 2019).

4. Projective Changes, Completeness, and Curvature

Quadratic curvature is central in the analysis of projective changes of Finsler metrics and their effect on completeness. Trivial projective changes of the form $F + d f$ preserve the geodesic foliation, if the function $D^+ + D^-$ (sum of forward and backward distances from a basepoint) is proper. Smooth mollifications and Lipschitz estimates ensure that the positive-definite quadratic forms (from Hessians or metric tensors) survive under this transformation, enabling control over completeness and curvature invariants.

In Lorentzian geometry, the completeness of induced Randers metrics (a Finsler class tightly tied to quadratic curvature) is equivalent to the global hyperbolicity of the spacetime, connecting quadratic curvature traits with causal and metric properties in a relativistic context (Matveev, 2011).

5. Extensions, Limitations, and Rigidity

The real-analyticity assumption is essential for the quadratic curvature classification on surfaces of negative Euler characteristic; the result fails in the smooth setting, where counterexamples abound. In higher dimensions, quadratic curvature invariants generate additional constraints and complexities—more independent first integrals and fibre Hessians must be considered, complicating classification and rigidity results. In Riemannian settings, the quadratic structure ensures that analogous classification can be achieved without analytic constraints, highlighting the special role of quadratic curvature in Finsler vs. Riemannian theory (Lang, 2019).

6. Connections to Flag Curvature and Projectively Flat Metrics

Quadratic curvature properties govern the projective flatness and flag curvature in Finsler and related metrics. Projectively flat, complete (α,β)-metrics are classified according to quadratic ODEs satisfied by the deformation functions, yielding metrics on spheres ($S^n$) projectively equivalent to standard metrics of constant sectional curvature. The flag curvature, scalar or tensorial, can be expressed in terms of such quadratic invariants and is preserved under projectively equivalent deformations when those deformations correspond to closed quadratic forms (closed 1-forms, closed conformal deformations) (Yang, 2015).

7. Summary Table: Quadratic Curvature Principles in Projective Equivalence

Setting	Structural Principle	Key Outcomes
Surfaces, χ < 0	Hessian proportionality	Classification via scaling + 1-form
Volume integrals on SM	Characteristic polynomial	Universal first integrals (quadratic)
Finsler metric changes	Properness of D⁺+D⁻; Lipschitz	Completeness via quadratic estimates
Entropy considerations	Integrability ↔ constancy	Rigidity from quadratic invariants
(α,β)-metrics on Sⁿ	ODEs for deformation functions	Flag curvature determined quadratically
Higher dimensions	More quadratic invariants	Complexity in classification

These quadratic curvature properties unify integrability, projective equivalence, metric completeness, and geometric entropy in the paper of Finsler and related geometries, providing a structural backbone for the classification, invariance, and rigidity phenomena in analytic and smooth settings.