Geometric and Topological Metrics

Updated 17 July 2025

Geometric and topological metrics are quantitative tools that measure distances, curvatures, and connectivity in mathematical, physical, and data-driven contexts.
They extend classical metric spaces by incorporating higher-order, ultrametric, and persistence-based approaches to capture complex structures.
These metrics enable robust analysis in areas such as quantum systems, digital imaging, and shape analysis by linking local geometry with global topology.

Geometric and topological metrics are foundational concepts in mathematics, physics, computer science, and data science, used to quantify, compare, and reveal structure in spaces, shapes, data sets, and physical systems. Their paper encompasses classical metric spaces, generalized higher-order metrics, the interaction of topology and geometry, and a broad range of applications from quantum many-body systems to material defects and digital imagery.

1. Classical and Generalized Metrics: Definitions and Structures

A metric space $(X, d)$ is a set equipped with a distance function $d(x, y)$ satisfying non-negativity, identity, symmetry, and the triangle inequality. This classical pairwise metric formalism captures geometric relationships between points and enables notions such as completeness, compactness, and curvature.

Generalizations expand this framework:

$k$ -metrics and strong $k$ -metrics: For $k \geq 2$ , a $k$ -metric is a real-valued function $d(x_1,\ldots,x_k)$ that, for $k=2$ , recovers a standard metric but for $k>2$ replaces the triangle inequality with a simplex inequality:

$d(x_1, \ldots, x_k) \leq \sum_{i=1}^{k} d(x_1,\ldots,x_{i-1}, y, x_{i+1},\ldots, x_k)$

Strong $k$ -metrics additionally satisfy a topological consistency condition ensuring their behavior is compatible with desirable continuity and embeddability properties (Barkan-Vered et al., 2023).

Ultrametrics and ultranorms: Metrics satisfying a strengthened triangle inequality, $d(x,y)\leq \max\{d(x,z),d(z,y)\}$ , yield spaces with strong separation and lead to totally disconnected topologies and topological dimension 0 (1503.02071).
Metric pairs and diagram spaces: In topological data analysis, the space of persistence diagrams can be metrized via Wasserstein-type distances, making the space of diagrams itself a metric space with structure inherited from the underlying ambient space (Che et al., 2021).

These generalized metrics support the development of new embeddings, such as higher-order Fréchet and $\ell_1$ embeddings for strong $k$ -metrics and hypertree metrics, thus generalizing classical results from finite metric embedding theory.

2. Topological Invariants and Their Relation to Metrics

Topological metrics measure "shape" and "connectivity" in a manner robust to continuous deformations, often mapping geometric or combinatorial input to invariants such as Betti numbers, Euler characteristic, Lusternik–Schnirelman category, or the ranks of persistent homology groups.

Euler characteristic ( $\chi$ ): A fundamental topological invariant, calculable for digital images and continuous spaces, serves as a quantitative measure of connectivity and holes (Boxer, 2021, Abaach et al., 2023).
Persistent homology and diagrams: Persistent homology captures the "lifespans" of topological features across filtrations, enabling comparison via metrics such as the $p$ -Wasserstein or bottleneck (infinity) distances between persistence diagrams. The geometry of these diagram spaces—completeness, geodesicity, and curvature—has been rigorously analyzed, revealing, for example, that diagram spaces with $p=2$ are Alexandrov spaces of nonnegative curvature and possess infinite Hausdorff and Assouad dimensions (Che et al., 2021).
Composite metrics: Practical tasks frequently combine geometric and topological metrics to capture both spatial and structural similarity. For instance, the sum of the Hausdorff distance, the difference in Euler characteristic, and other topological or geometric pseudometrics provides a more discriminating composite metric for digital topology and image analysis (Boxer, 2021).

3. Geometry–Topology Interplay: Hyperconvexity, Curvature, and Filtrations

A central modern theme is the interplay between geometric properties (like curvature) and topological invariants, often realized through ball intersection patterns and persistent homology.

Hyperconvexity: A (metric) space is hyperconvex if every collection of closed balls with pairwise-overlapping radii $r_i + r_j \geq d(x_i, x_j)$ has nonempty total intersection. This property has topological consequences: hyperconvex spaces are contractible and have trivial higher-dimensional Čech persistent homology—a signature of "flat" or "tripod" geometry (Joharinad et al., 2020, Joharinad et al., 2022).
Curvature via ball intersections: Recent developments define curvature not via classical differential geometry but through the minimal scaling needed for all relevant balls to intersect, generalizing midpoints and circumcenters (tripod spaces) (Joharinad et al., 2022). In non-positive-curvature (NPC) spaces, the intersection scale is no greater than in the Euclidean comparison triangle.
Čech and Vietoris–Rips filtrations: The process of growing balls around points and recording overlaps builds topological filtrations whose homological features (captured by persistent homology) encode both geometric (e.g., curvature) and topological (e.g., connectivity) information (Joharinad et al., 2020).

4. Metrics in Specific Mathematical and Physical Settings

A range of applications illustrate how geometric and topological metrics function as analytical tools across domains:

Quantum many-body and topological order: In the paper of quantum systems (such as the toric code), geometric entanglement—measured by the logarithm of the overlap between a state and the closest product state—splits into a non-universal "bulk" (boundary law) component and a universal, topological component equating to the topological entanglement entropy. This affirms the utility of multipartite geometric entanglement as a topological order parameter robust under perturbations (1108.1537).
Shape analysis and elastic graphs: For complex networks (neuronal trees, vasculature), elastic shape analysis frameworks use quotient structures and elastic metrics on curves/edges to define geodesics, means, and statistical summaries in shape space. Registrations of graphs via permutation group actions, together with elastic metrics, provide invertible, quantitative shape comparisons that general topological invariants can't offer (Guo et al., 2020).
Symmetry and minimal metrics on groups: In topological group theory, a minimal metric is a left-invariant metric that induces a canonical local (and sometimes global) Lipschitz geometry, playing a pivotal role in the solution of Hilbert’s fifth problem and in the classification of groups admitting Lie group structures (1611.04057).
Geometric and topological features in images: In digital topology, the Hausdorff metric may fail to distinguish fundamentally different shapes. Supplementing with (pseudo-)metrics reflecting convexity deviations, Euler characteristic differences, and set diameters improves both the geometric and topological fidelity of similarity measures (Boxer, 2021). In image analysis and classification, combining topological data analysis (persistent homology) features with geometric measures such as Lipschitz–Killing curvatures yields improved performance and interpretability in biomedical classification tasks (Abaach et al., 2023).
Defects, disclinations, and self-force in materials: In the context of elastic media and gravitational analogs, curvature induced by topological defects (disclination quadrupoles) is captured via a conformal metric. The associated Poisson equation encodes defect-induced curvature, with different geometric arrangements (linear vs. square quadrupoles) giving rise to characteristic angular harmonics in the curvature field, which in turn govern self-energy and self-force phenomena (Carvalho et al., 29 Apr 2025).
Black holes and spacetime metrics: In general relativity, the spherically symmetric black hole metric

$ds^2 = -f(r) dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta d\phi^2)$

reflects both the local geometry and the global topology of the event horizon, which enters thermodynamic quantities such as entropy and phase transitions. Topological invariants like the Euler characteristic, and geometric invariants via the Gauss–Bonnet formula, directly relate to observable and theoretical properties, including uniqueness and phase structure when matter fields (such as those causing Bose–Einstein condensation) are present (Chen, 5 Oct 2024).

5. Computational and Algorithmic Perspectives

Several computational strategies and algorithmic innovations have emerged to operationalize geometric and topological metrics:

Construction sequence and geometric realizability algorithms: For incidence configurations, algorithms leveraging projective bases and free-parameter elimination can enumerate all combinatorial or topological configurations and test for geometric realizability by reducing the problem to the existential theory of the reals (1309.3201).
Encounter graph metrics and mapping by swarms: In distributed robotics and swarm mapping, coordinate-free local interactions between moving agents give rise to encounter graphs. Edge weights defined from encounter times enable the reconstruction of geometric metrics and the extraction of topological invariants through manifold learning and persistent homology under rigorous convergence and stability conditions (1607.00051).
Average-pooling for topological smoothing: In machine learning for 3D biomedical images, average-pooling-based morphological smoothing, informed by geometric estimates of tubular radii, enables parameter-free, topology-preserving segmentation and centerline detection, as reflected in improved Betti error rates and clDice scores for axon tracing tasks (Shamsi et al., 2023).
Metric comparisons and topology induction: Studies of metric topologies over classes of regions—convex, unions, star-shaped—demonstrate that, for convex regions, natural metrics (Hausdorff, Wasserstein, symmetric difference, etc.) induce the same topology, while for more general regions, relations between these metrics become strictly ordered or even incomparable, highlighting the plurality of “reasonable” metrics in more complex settings (Davis, 2021).

6. Emerging Themes and Future Directions

Integration of geometric and topological features: Across scientific disciplines, combining geometric quantification (e.g., curvature, area, mean radius) and topological invariants (e.g., Betti numbers, persistence diagrams) has led to more discriminating and informative metrics for data analysis, recognition, and classification (Abaach et al., 2023).
Quantitative curvature and scaling in persistent homology: Recent work analyzes curvature not through the lens of smooth Riemannian geometry but via scaling properties of ball intersections and filtrations, enabling quantitative and even real-valued geometric invariants to supplement or refine classic integer-valued topological summaries (Joharinad et al., 2020, Joharinad et al., 2022).
Generalized distance relationships: The development of $k$ -metrics and associated embedding theorems suggests new invariants and tools for capturing higher-order geometric structure and may catalyze advances in computational topology, combinatorics, and geometric group theory (Barkan-Vered et al., 2023).
Interaction with physics and materials science: Geometric and topological metrics form the basis for understanding phenomena such as topological order in quantum systems, defect-induced curvature in elastic materials, and phase transitions in black holes. These insights connect local geometric measurements to global topological classifications and physical observables (1108.1537, Carvalho et al., 29 Apr 2025, Chen, 5 Oct 2024).
Robust data analysis techniques: Persistent homology and the geometry of diagram spaces, with rigorously proven metric properties and stability results, underlie the foundation for practical, noise-robust topological data analysis in high-dimensional settings (Che et al., 2021).

In sum, geometric and topological metrics not only provide the mathematical infrastructure for measuring and comparing diverse objects across pure and applied fields but also facilitate deep connections between local geometric phenomena and global topological structure. Modern research shows the power of these metrics in both abstract theory and concrete computational applications—shaping our understanding of spaces, shapes, data, and physical systems.