Generalized Projective Riemann Curvature

• Generalized projective Riemann curvature is a family of invariant curvature tensors defined under projective transformations that unifies approaches in Riemannian and Finsler geometries.
• It emphasizes the role of projective invariance in classifying manifolds and analyzing geodesic flows through conditions such as semisymmetry and pseudosymmetry.
• This framework provides practical insights into curvature rigidity, quadratic curvature properties, and metric classifications across various geometric settings.

The concept of generalized projective Riemann curvature encompasses a family of curvature tensors and associated geometric structures designed to encode those aspects of curvature that remain invariant under projective (geodesic-preserving) transformations. Originating in classical Riemannian geometry as the projective curvature tensor, and extended substantially in Finsler and spray geometry, these invariants capture a fundamentally projective notion of curvature, with deep implications for the classification of manifolds, the structure of geodesic flows, and the theory of pseudo-symmetric geometric structures. The recent literature has developed new frameworks for these generalized projective invariants, particularly in Finsler geometry, highlighting quadratic curvature properties and establishing a unified language for different signature settings (Shaikh et al., 2016, Sadeghzadeh et al., 23 Nov 2025).

1. Projective Curvature in Riemannian and Semi-Riemannian Geometry

On a semi-Riemannian manifold $(M^n, g)$ with Levi-Civita connection $\nabla$, the Riemann curvature tensor $R_{ijkl}$ is the central curvature invariant. The projective curvature tensor $P_{ijkl}$ is defined by

$$P_{ijkl} = R_{ijkl} - \frac{1}{n-1} (S_{jk}g_{il} - S_{ik}g_{jl}),$$

where $S_{ij}$ is the Ricci tensor $R^k{}_{ikj}$, and $g_{ij}$ is the metric (Shaikh et al., 2016). The tensor $P$ is the unique part of $R$ invariant under geodesic-preserving (projective) transformations of the connection.

$P$ fails to satisfy all the algebraic symmetries (notably the pair symmetry $D_{ijkl}=D_{klij}$) of generalized curvature tensors, which distinguishes it fundamentally from the Riemann, Weyl, or conformal curvature tensors. This lack of full symmetry has direct geometric consequences, affecting the types of curvature restrictions and invariants possible.

2. Semisymmetric and Pseudosymmetric Structures Induced by Projective Curvature

Classical curvature-restricted structures—semisymmetric and pseudosymmetric manifolds—are defined via identities involving operators such as the Kulkarni–Nomizu product and curvature derivatives. In the context of $P$, these become "projective-semisymmetric" ($P\cdot P=0$) and "projective-pseudosymmetric" ($P\cdot P = L Q(g, P)$, for some $L\in\mathbb{R}$), where $Q(A,B)$ denotes the Kulkarni–Nomizu product of symmetric $(0,2)$ tensors $A$ and $B$ (Shaikh et al., 2016).

The behavior of these structures under projective transformations is distinct from that in the generalized tensor case. For instance, $R\cdot P+$cyclic $=0$ holds only if the manifold is Ricci semisymmetric ($R\cdot S=0$), and $P\cdot P+$cyclic $=0$ often forces algebraic conditions that force the Einstein property in Riemannian signature.

3. Generalized Projective Curvature in Finsler Geometry

Finsler geometry generalizes projective invariance through the notion of projective sprays, Jacobi endomorphisms, and projective curvatures defined directly on tangent bundles (Sadeghzadeh et al., 23 Nov 2025, Bucataru, 2014). Let $F: TM \to [0,\infty)$ be a Finsler metric and $G^i(x, y)$ its geodesic spray coefficients. Two metrics are projectively related if $G^i(x, y) = \bar{G}^i(x, y) + P(x, y) y^i$, for some 1-homogeneous function $P$.

The classical projective Riemann curvature can be expressed for a projective spray

$PG^i(x, y) = G^i(x, y) - \rho_G(x, y) y^i,$

with $\rho_G$ a 1-homogeneous function satisfying $\rho_{\tilde G} = \rho_G + P$ under $G \mapsto \tilde G = G + P y$. The projective Riemann tensor derived from the curvature of $PG$ is then invariant under such changes.

Generalized projective Riemann curvature is this invariant curvature structure for an arbitrary compatible choice of $\rho_G$, unifying previous formulations and admitting new curvature identities and classification patterns (Sadeghzadeh et al., 23 Nov 2025). Principal cases include volume-derived choices ($\rho_G = S/(n+1)$, with $S$ the S-curvature), twisted by 1-homogeneous invariants, and directional invariants such as $\rho_G = \mathcal{D}V$, where $V$ is directionally invariant and $\mathcal{D}V = (\ln V)_{.r}G^r$.

4. Quadratic Curvature Properties and Metric Classifications

Within this framework, curvature quadraticity is defined relative to the projective spray. A Finsler metric is said to be "generalized projectively $R$-quadratic" (GPR-quadratic) if the Berwald curvature of $PG$, denoted $$PB_j{}^i{}_{kl} = \frac{\partial^3 PG^i}{\partial y^j \partial y^k \partial y^l}$$, satisfies

$$PB_j{}^i{}_{kl|0} - (\rho_G)_{.r} PB_j{}^r{}_{kl} y^i = 0,$$

where $|$ denotes the horizontal covariant derivative and $.r$ vertical differentiation (Sadeghzadeh et al., 23 Nov 2025). For specific $\rho_G$, this reduces to known quadratic curvature conditions, such as the Douglas condition for $\rho_G = -S/(n+1)$.

This characterization produces a spectrum of metrics: Berwald metrics (all $PB = 0$) are always GPR-quadratic, as are projectively flat metrics.

5. Rigidity and Projective Invariants in Spray Geometry

In spray geometry and Finsler flag curvature theory, generalized projective Riemann curvature provides an obstruction to the existence of projectively related Finsler metrics with identical curvature tensors. For Finsler metrics of non-vanishing scalar flag curvature, rigidity holds: no nontrivial projective deformation (by a Funk function) produces another Finsler-metrizable spray with the same projective Riemann curvature (Bucataru, 2014). When the scalar flag curvature vanishes, there exist infinitely many such metrics—a phenomenon connected to Hilbert's Fourth Problem.

The transformation law for the Jacobi endomorphism (Riemann curvature) under a projective change in the spray can be stated as

$$\widehat{\Phi} = \Phi + (P^2 - S(P)) J - [d_J(S(P)-P^2) + 3(P\,d_J P - d_h P)] \otimes C,$$

where $\Phi$ and $\widehat{\Phi}$ are the respective Jacobi endomorphisms, $J$ is the vertical endomorphism, $C$ is the Liouville vector, and $d_h, d_J$ are the Frölicher–Nijenhuis derivations (Bucataru, 2014).

6. Characterization Theorems, Examples, and Extensions

Several main results trace the implications of projective curvature constraints:

• If a Riemannian manifold satisfies projective semisymmetry or certain pseudosymmetric identities, it must be Einstein (Shaikh et al., 2016).
• In the Finsler context, explicit constructions exhibit the strictly projective nature of certain sprays and demonstrate the lack of metrizable representatives with prescribed curvature invariants outside special cases (Sadeghzadeh et al., 23 Nov 2025, Bucataru, 2014).

Illustrative examples include:

• Metrics that are Ricci semisymmetric but not projectively semisymmetric, showing sharp distinctions between Riemann and projective curvature restrictions.
• Warped and conformally flat metrics with non-Einstein projective pseudosymmetry.
• Venzi spaces based on $P$, with the property that non-null invariants force constant curvature.

7. Geometric and Analytical Significance

Generalized projective Riemann curvature provides a powerful language for analyzing intrinsic properties of geometric structures under projective equivalence. By replacing volume-based invariants with arbitrary 1-homogeneous functions or directional data, it expands the toolkit for both classification and the analysis of geometric flows in anisotropic and indefinite signature settings (Sadeghzadeh et al., 23 Nov 2025). Its connections to curvature rigidity, quadratic conditions, and invariants under transformation underpin a substantial portion of modern work in geometric analysis and mathematical physics.

These structures also delineate the landscape of projective classes: "flat" projective classes (vanishing flag curvature) are rich in Finsler solutions, while curved cases are severely rigidified by the projective curvature constraints (Bucataru, 2014). The resulting projective invariants unify and extend many classical curvature theories, with ongoing research probing their analytic, cohomological, and deformation-theoretic implications.