Overview of Discriminant Riemannian Metrics

• Discriminant Riemannian metrics are generalized metrics that incorporate algebraic and geometric degeneracies, enabling the study of signature changes and controlled metric behavior.
• They utilize precise local normal forms and analytic techniques to manage degeneracies along designated submanifolds, ensuring the uniqueness of geodesics and coherent topological structure.
• These metrics underpin applications such as Riemannian discriminant analysis, pseudo-Riemannian modeling, and holonomy reduction, yielding improved performance in high-dimensional optimization.

A discriminant Riemannian metric is a generalization of the classical Riemannian metric that incorporates algebraic or geometric degeneracy, either globally via tensor invariants or locally along submanifolds known as discriminant loci. These metrics arise in diverse contexts: subspace optimization on manifolds with Fisher-type criteria, degenerations along analytic divisors, sudden signature changes in pseudo-Riemannian geometry, and the realization of algebraic invariants—such as the discriminant of a cubic polynomial—as curvature tensors in high-dimensional manifold theory. Their study involves advanced techniques from Riemannian manifold optimization, singularity theory, pseudo-Riemannian geometry, and symmetric space theory.

1. Algebraic and Geometric Definitions

Discriminant Riemannian metrics encompass several precise definitions rooted in the degeneration, reduction, or specialization of the metric tensor:

• ODD metrics: On a real-analytic manifold $M$, a symmetric semipositive-definite tensor $g$ is called an ODD (Orthogonally Degenerating on a Divisor) metric if it degenerates only along a finite union of analytic submanifolds $\mathcal{D}$ ("discriminant divisor") and, recursively, its restriction to each component is again an ODD metric (Braun, 2022). The local normal form near a general point of the divisor is $g = \sum_{i=1}^{n-1} dx_i^2 + y\, dy^2$.
• Signature-changing discriminant metrics: On surfaces, a pseudo-Riemannian metric can be written

$$ds^2 = a(x,y)\, dx^2 + 2b(x,y)\, dx\, dy + c(x,y)\, dy^2,$$

where the discriminant function $\Delta(x,y) = a c - b^2$ vanishes along a smooth curve $D$. At points on $D$, the metric changes from Riemannian ($\Delta > 0$) to Lorentzian ($\Delta < 0$), and geodesic flows exhibit singularities classified according to contact types and cubic equation roots (Remizov et al., 2015).

• Cubic discriminant tensors on 8-manifolds: In eight dimensions, a "cubic discriminant" is a symmetric rank-4 tensor field $\Phi \in \Gamma(S^4 T^*M)$ invariant under $SO(4)_{ir}\subset GL(8)$, the image of the irreducible representation of $SO(4)$ (Hristova et al., 16 Aug 2025). Such a tensor is linearly equivalent at each point to the classical discriminant of a cubic polynomial, and its parallelism (integrability) restricts the metric to special Wolf spaces ($G_2/SO(4)$, $G_{2(2)}/SO(4)$) or the flat case.

2. Analytical Structures and Metric Degeneracies

Discriminant metrics require refined analysis of degeneracies and their compatibility:

• Inductive degeneracy structure: The metric is nondegenerate off $\mathcal{D}$ and degenerates strictly orthogonally on each component. The degenerate locus does not interfere with metric-space topology: the induced metric $d_g$ continues to generate the manifold topology and admits unique geodesics even at generic points of $\mathcal{D}$ (Braun, 2022).
• Local normal forms for signature change and discriminant degeneracies are classified by contact order and tangency of isotropic directions, with explicit coordinates ensuring analytic compatibility (no infinitely flat remainders).
• Orthogonal frames and connections: Away from degeneracy, Gram–Schmidt yields analytic orthonormal frames. At degeneracy, flat/sharp maps and Christoffel symbols become meromorphic but remain valid on the sheaf of algebraic vector fields. The Levi-Civita connection adapts, ensuring existence and uniqueness of ODD geodesics at generic points even with degeneracy.

3. Discriminant Metrics in Subspace Learning and Optimization

Discriminant Riemannian metrics underpin advanced algorithms in subspace learning:

• Riemannian Discriminant Analysis (RDA): Linear discriminant analysis (LDA), traditionally Euclidean, is lifted to a Riemannian setting on the Stiefel or Grassmann manifold

$$\mathrm{St}(D,d) = \{U \in \mathbb{R}^{D \times d}: U^T U = I_d\}.$$

The optimization problem becomes

$$f(U) = \mathrm{tr}(U^T S_W U) - \mathrm{tr}(U^T S_B U),$$

with closed-form Riemannian gradient and Hessian, and second-order trust-region algorithms for globally convergent subspace learning (Yin et al., 2021).

• Empirical superiority: Across multiple datasets (COIL-20, CMU-PIE, MNIST, etc.), RDA achieves clustering accuracies and classification rates well above prior Euclidean and Riemannian methods, sometimes outperforming by 10–40 points in accuracy. The subtraction-form Fisher criterion avoids matrix inversion, improving numerical stability and the recovery of optimal subspaces.
• Sparsity regularization: An $\ell_1$-penalty ($f(U)+\lambda \|U\|_1$) can be incorporated, yielding further improvements in high-dimensional embedding accuracy.

4. Singularities and Signature Change Along Discriminant Loci

The geometric and dynamical consequences of discriminant metrics are exemplified in the pseudo-Riemannian context:

• Classification of geodesic-flow singularities: Points where the metric changes signature (along $\Delta=0$) are loci of nonstandard geodesic behavior, governed by cubic root structures in the admissibility equation $M(q,p)=0$. The contact order (transverse, first-order tangency, etc.) produces distinct local normal forms and phase portraits (Remizov et al., 2015).
• Applications: Discriminant metrics model signature change in relativity, impact billiard-type dynamics in Minkowski space, and relate to Gauss–Bonnet-type index theorems and the study of umbilic points.

5. Cubic Discriminants, Holonomy Reduction, and Classification in High Dimensions

A specialized algebraic perspective emerges from cubic discriminant tensors and their geometric implications:

• Reduction to $SO(4)_{ir}$ structures: On 8-manifolds, the existence of a parallel cubic discriminant tensor $\Phi$ entails a reduction of the holonomy group to $SO(4)_{ir}$, further imposing an almost quaternion-Hermitian structure and specialized curvature decompositions (Hristova et al., 16 Aug 2025).
• Classification theorem: Integrable cubic discriminant metrics are uniquely realized by the quaternion-Kähler Wolf spaces $G_2/SO(4)$ (compact) and $G_{2(2)}/SO(4)$ (noncompact), as well as flat $\mathbb{R}^8$. The curvature tensor splits into "hyper-Kähler" and "projective" components, with the discriminant entering as a quartic curvature invariant.
• Curvature-based characterization: The existence and parallelism of a cubic discriminant tensor at each point dictate that the metric is locally isometric to the exceptional symmetric spaces cited above.

6. Obstructions, Duality, and Distinguished Geometric Structures

Discriminant metrics provide a framework for analyzing obstructions to distinguished curvature conditions:

• Lorentzian–Riemannian duality: A distinguished Riemannian metric (e.g., Ricci-flat or locally symmetric) can be probed via its Lorentzian dual. Penrose's plane-wave limit reduces the analytic complexity to the Hessian of a single potential function, whose properties encode necessary local obstructions (2207.14701).
• Global obstructions: Using Wick rotation and Bochner-type arguments, one rules out compact Riemannian manifolds that are deformations of constant-curvature spaces along closed vector fields except in the flat case.
• Analytic techniques: Null coordinates, scaling limits, and Rosen/Brinkmann coordinate forms yield criteria under which geodesics, Ricci-parallelism, and local symmetry cannot be achieved, governed by simple ODE or PDE conditions on the Penrose potential.

7. Current Directions and Implications

Discriminant Riemannian metrics unify phenomena across geometry, optimization, and mathematical physics. Their versatility lies in encoding algebraic invariants, managing controlled degeneracy, and structuring high-dimensional geometric subspaces. Active research examines loosening regularity conditions (beyond closed vector fields), exploiting cubic discriminant structures for new holonomy reductions, and extending analysis to metrics with special algebraic properties (Bach-flatness, scalar-flatness) and to broader classes of pseudo-Riemannian geometries.

This synthesis highlights the foundational impact and multifaceted applications of discriminant Riemannian metrics, encompassing optimization on matrix manifolds, singularity theory, structure reduction, and novel geometric invariants.