Singular Riemannian Metrics

• Singular Riemannian Metrics are symmetric two-tensors on manifolds that lose non-degeneracy or smoothness on designated subsets, extending classical metric theory.
• Regularization methods like the radical-annihilator framework and Ricci–DeTurck flow approximation enable effective analysis of their spectral, curvature, and embedding properties.
• Model examples—including sub-Riemannian, lightlike, and analytic degenerate metrics—illustrate diverse applications and challenges in modern geometric analysis.

A singular Riemannian metric is a symmetric two-tensor on a differentiable manifold that loses non-degeneracy or smoothness on a closed subset, a singular set, or along certain directions. These metrics appear in several contexts: geometric analysis on stratified spaces, sub-Riemannian and lightlike (degenerate) structures, and in singularity formation in flows and embedding problems. Their study involves analytic, algebraic, and geometric challenges, including breakdown of classical invariants, the need for generalized spectral/curvature theories, and new rigidity/regularization phenomena.

1. Definitions and Types of Singular Riemannian Metrics

A classical Riemannian metric $g$ is a smooth, positive-definite symmetric two-tensor. A singular Riemannian metric refers to a tensor field with at least one of the following pathologies:

• It is only measurable or $L^\infty$, possibly with positive-definite values only almost everywhere, and may be smooth only off a singular set $S$.
• It is smooth but fails to be non-degenerate everywhere, i.e., $\det g$ vanishes on a closed set or along a subbundle (e.g., lightlike or sub-Riemannian metrics).
• It blows up or degenerates in controlled fashion near stratified subsets (e.g., $g = dx^2 + x^{-\beta}g_1(x)$ near $x=0$).

Characteristic examples:

• Sub-Riemannian (Carnot–Carathéodory) metrics: smooth quadratic forms $h$ on a distribution $D\subset TM$ of rank $k<n$; length is infinite on $TM\setminus D$.
• Lightlike metrics: symmetric bilinear forms on $TM$ of corank 1 (positive semi-definite, not definite) (Bekkara et al., 2011).
• Analytic metrics that are degenerate at isolated points, such as $g_{nn}(x) = (|x'|^2 + x_n^{2\ell})F_0(x)$ in a neighborhood of the origin (Enciso et al., 2020).
• Measurable metrics $g\in L^\infty$ smooth except on $S\subset\mathbb R^n$ of finite Minkowski codimension, with scalar curvature constraints (Burkhardt-Guim, 2024).

2. Geometric Structure and Regularization

The loss of non-degeneracy or regularity invalidates standard geometric constructions such as the Levi-Civita connection or curvature. Several frameworks and regularization schemes have been introduced:

• Radical and Radical-Annihilator: For a degenerate metric $g$, the radical at $p$ is $\{v\in T_pM: g_p(v,w)=0\,\forall w\}$; the dual radical-annihilator consists of covectors vanishing on the radical. Operations such as contraction or covariant differentiation must be restricted to radical-annihilator slots (Stoica, 2011).
• Covariant Derivatives and Curvature: On radical-stationary and semi-regular manifolds (where certain forms and derivatives are radical-annihilator), one defines a lower covariant derivative and constructs a Riemann curvature with classical symmetries. The standard Einstein tensor blows up as $\det g\to0$, but the densitized version (weight-two density $\widetilde G$) remains smooth, enabling extension of general relativity through signature change (Stoica, 2011).
• Pseudometric Reductions: In Lie algebroid/information geometry frameworks, degenerate symmetric forms arising from contrast functions induce a well-defined non-degenerate (pseudo-)metric on the quotient by the kernel. The generalized Levi-Civita connection descends to the transverse bundle (Grabowska et al., 2020).
• Approximation and Smoothing: $L^\infty$ metrics close to Euclidean, smooth outside $S$ with nonnegative scalar curvature, may be approximated by smooth metrics via Ricci–DeTurck flow, so that the flow converges locally to $g$ away from $S$ and nonnegative scalar curvature is preserved under appropriate Minkowski content conditions (Burkhardt-Guim, 2024). Sub-Riemannian structures can be approximated by a family of smooth Riemannian metrics degenerating in controlled fashion along the higher-step directions, yielding Hausdorff and spectral convergence to the limit structure (Harakeh et al., 8 Dec 2025).

3. Singularity Classes and Model Geometries

Prominent classes, stratified by their singular set and degeneracy type, include:

Setting	Metric Form / Description	Singular Set
Sub-Riemannian	$g$ positive-definite on $D \subset TM$, infinite elsewhere	$TM \setminus D$
Lightlike (Null)	$g$ positive semidefinite, corank 1; $g$ on $N \subset TM$ zero	Characteristic line $N$
Analytic Isolated Zero	$g$ analytic, $\det g = 0$ only at one point (e.g., $|x'|^2 + x_n^{2\ell}=0$)	$(0,0)$
Stratified/Boundary	$g = dx^2 + x^{-\beta}g_1(x)$, blows up as $x \to 0$	Boundary $x=0$

Further, the notion of generalized smooth distributions (i.e., modules of vector fields of possibly varying rank) accommodates fiberwise singular metrics and induces associated Laplace-type operators (Androulidakis et al., 2018).

4. Spectral Geometry and Weyl Laws

Spectral analysis of singular metrics reveals departures from classical asymptotics. For metrics blowing up near a stratified boundary or singular submanifold, spectral invariants are modified:

• Weyl's Law for Singular Manifolds: For $M^n$ with singularity modeled in a collar as $g = dx^2 + h(x)$, assuming curvature and injectivity radius blow up like $O(x^{-2})$ and $O(x)$ respectively, and with "slowly varying" regularized volume function $V(\lambda)$, the spectral counting function $N(\lambda)$ obeys

$$N(\lambda) \sim \frac{\omega_n}{(2\pi)^n}\,\lambda^{n/2}\,V(\lambda)$$

as $\lambda \to \infty$ (Chitour et al., 2019). The function $V(\lambda)$ can exhibit logarithmic or iterated log growth, exponential in $(\log \lambda)^\alpha$, etc., and hence produces non-classical spectral exponents.

• Critical and Supercritical Cases: For the model $g = dx^2 + x^{-\beta} g_1(x)$ on $[0,1]\times M$ near boundary,
  • Subcritical ($\beta < 2/n$): classical Weyl law holds.
  • Critical ($\beta = 2/n$): $N(\lambda) \sim C\,\lambda^{(n+1)/2} \log \lambda$; eigenfunction mass is distributed across all $x \in [\lambda^{-1/2},1]$ scales, not confined to a boundary layer (Dietze, 27 Oct 2025).
  • Supercritical ($\beta > 2/n$): $N(\lambda) \sim C\,\lambda^{d/2}$ ($d = n(1+\beta/2)$), and eigenfunction $L^2$ mass concentrates sharply in $x \sim \lambda^{-1/2}$ boundary layers (Dietze, 23 Jan 2026).
• Concentration/Delocalization: For metrics with power-type singularities, average eigenfunction densities concentrate at the singular boundary as $\lambda\to\infty$, with explicit rates in the Wasserstein distance for various $p$. In the critical regime, the concentration is uniform over log-scales of $x$ near the boundary (Dietze, 23 Jan 2026, Dietze, 27 Oct 2025).
• Generalized Distributions: For singular metrics on generalized distributions of non-constant rank, Laplacians are hypoelliptic and essentially self-adjoint; spectral theory develops via the longitudinal pseudodifferential calculus of the smallest singular foliation containing the distribution (Androulidakis et al., 2018).

5. Algebraic and Analytic Embedding Results

Classical isometric embedding theorems break down for singular metrics at degenerate points. Using ramified versions of the Cauchy-Kovalevskaya theory (Leray, Choquet-Bruhat), local analytic embeddings are recovered on finite branched covers, increasing the required codimension:

• For an analytic metric $g$ degenerate at $0\in\mathbb{R}^n$, taking normal forms where $g_{nn}(x) = (|x'|^2 + x_n^{2\ell})F_0(x)$, there exists a branched cover $\Pi:U'\to U\setminus\{0\}$ such that $(U',\Pi^*g)$ admits an analytic isometric embedding into $\mathbb{E}^{(n^2+3n-4)/2}$ (Enciso et al., 2020).

This generalizes Cartan-Janet's theorem, where codimension jumps by $n-2$ due to the singularity.

6. Rigidity, Sub-Rigidity, and Singular Structures

Classical geometric rigidity, as in Cartan's finite-type and Gromov's $k$-rigidity, fails for singular metrics, but refined sub-rigidity phenomena occur:

• Sub-Riemannian Contact Structures: Contact sub-Riemannian metrics on $(2d{+}1)$-manifolds are $(4,1)$-sub-rigid: local 4-isometries with trivial 1-jet at a point automatically have trivial 2-jet (Bekkara et al., 2011).
• Generic Lightlike Metrics: For $n\geq4$, generic lightlike metrics are $(3,1)$-sub-rigid: 3-infinitesimal isometries with trivial 1-jet are 2-jet trivial.

Isometry pseudogroups can be infinite-dimensional in the integrable (leafwise) or transversally Riemannian case; finite-dimensional in the presence of contact (non-integrable) or genericity constraints.

7. Curvature and Flow in the Singular Setting

Classical curvature tensors are unavailable for general singular metrics, but curvature invariants can be defined via measure theory and weak formulations:

• Ricci Curvature as a Measure: On $C^{1,1}$-manifolds with singular metrics in $W^{1,1}_{\text{loc}}$, Ricci curvature is defined as an $S^2(T^*M)\otimes D^*$-valued measure via an integral Bochner-type formula over vector-valued half-densities. Examples include cones (concentrated Ricci at the vertex), glued manifolds (boundary layer measures), and Kähler manifolds (currents proportional to $\partial\overline\partial \ln\det h$) (Lott, 2015).
• Weak Ricci Flow: There is a framework for flows starting from singular initial metrics, with finiteness and lower bound conditions formulated at the level of quadratic forms on vector-valued half-densities. Stability, compactness, and extension to singular initial times are established (Lott, 2015).
• Spectral and Heat Kernel Regularization: For singular or sub-Riemannian metrics approximated by smooth Riemannian families, Laplacians and volumes converge to Popp's volume and sub-Riemannian Laplacian, allowing the transfer of spectral and heat kernel results to the singular limit (Harakeh et al., 8 Dec 2025).

• "Weyl’s law for singular Riemannian manifolds" (Chitour et al., 2019)
• "Riemannian metrics and Laplacians for generalised smooth distributions" (Androulidakis et al., 2018)
• "Information geometry on groupoids: the case of singular metrics" (Grabowska et al., 2020)
• "Convergence speed for the average density of eigenfunctions for singular Riemannian manifolds" (Dietze, 23 Jan 2026)
• "The critical case for the concentration of eigenfunctions on singular Riemannian manifolds" (Dietze, 27 Oct 2025)
• "On Singular Semi-Riemannian Manifolds" (Stoica, 2011)
• "Singular Riemannian metrics, sub-rigidity vs rigidity" (Bekkara et al., 2011)
• "Ramified local isometric embeddings of singular Riemannian metrics" (Enciso et al., 2020)
• "Approximating sub-riemannian structures by Riemannian metrics and spectral convergence" (Harakeh et al., 8 Dec 2025)
• "Ricci measure for some singular Riemannian metrics" (Lott, 2015)
• "Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set" (Burkhardt-Guim, 2024)