Persistent Manifolds

• Persistent manifolds are geometric structures that remain stable under perturbations, uniting invariant manifold theory with persistent homological techniques.
• They employ methods such as Lyapunov–Perron constructions, graph transforms, and fixed-point strategies to maintain manifold invariants even in noncompact settings.
• Their applications span dynamical systems and data-driven manifold reconstruction, facilitating robust triangulation and multiparameter persistence analysis.

A persistent manifold is a manifold structure equipped with persistent or stability properties under perturbations, typically arising in dynamical systems, topological data analysis, or topological invariants extracted from geometric data. The concept incorporates both classical persistence in the sense of homological invariants that survive over parameter scales and the dynamical systems perspective of invariant manifolds persisting under system perturbation. A rigorous synthesis of these ideas requires precise definitions of the manifold class (e.g., smooth, Riemannian, compact/noncompact), the ambient geometric or dynamical context (e.g., bounded geometry, vector fields), and the technical framework for persistence (e.g., Lyapunov–Perron methods, persistent homology modules, sheaf-theoretic persistence). Persistent manifolds have emerged prominently in the theory of invariant manifolds for noncompact dynamical systems, multiparameter persistent homology, and as topological objects reconstructed via persistent triangulation from sampled data.

1. Definitions and Fundamental Notions

Several formulations of persistent manifolds exist, corresponding to different mathematical settings:

A. Persistent Normally Hyperbolic Invariant Manifolds (NHIMs)

For a complete, smooth Riemannian manifold $(Q,g)$ of bounded geometry and a $C^r$-flow $\Phi^t$ on $Q$, a connected submanifold $M \subset Q$ is a (possibly noncompact) NHIM if:

• Invariance: $\Phi^t(M) = M$, $\forall t \in \mathbb{R}$.
• Invariant Splitting: $TQ|_M = TM \oplus E^+ \oplus E^-$, with bounded, $D\Phi^t$-invariant projections.
• Exponential Contraction/Expansion:

$\|D\Phi^t|_{E^-}\| \leq C_- e^{-P_- t}$ ($t\geq 0$), $\|D\Phi^{-t}|_{E^+}\| \leq C_+ e^{-P_+ t}$ ($t\geq 0$), $\|D\Phi^t|_{TM}\| \leq C_M e^{P_M|t|}$ ($\forall t$).

• Spectral Gap: For $r$-normal hyperbolicity, $P_- < -r P_M$ and $r P_M < P_+$ for some $r > 1$.

These structures persist under small $C^1$-perturbations of the flow if $Q$ has bounded geometry, with the persistent manifold $M'$ remaining $C^{k,\alpha}$, diffeomorphic to $M$, and lying in a controlled neighborhood of $M$ (Eldering, 2012).

B. Persistent Triangulations and Data-Driven Manifolds

Given a compact, smooth $d$-manifold $M \subset \mathbb{R}^N$ of reach $\tau > 0$, a finite sample $X \subset M$ with Hausdorff distance $d_H(X, M) < \delta < \tau$ gives rise to a one-parameter family of simplicial complexes $\{K_\varepsilon\}$ (e.g., Vietoris–Rips, Čech, or witness complexes). For a critical $\varepsilon^*$ in the stability regime $2\delta < \varepsilon < \tau-\delta$, $K_{\varepsilon^*}$ is called an approximate triangulation of $M$ if $$H_*(K_{\varepsilon^*}; \mathbb{F}) \cong H_*(M; \mathbb{F})$$; these complexes persist in homology over relevant scale intervals (Knudson, 2020).

C. Persistent Manifolds in Multiparameter Persistence Theory

A persistent manifold may also mean a manifold $M$ along with a multiparameter filtration, e.g., via a family $\{\tilde f_t\}_{t \in [0,1]}$ of functions and the corresponding module $H_jF(a, b, c)$ indexed by $I \times I \times \mathbb{R}$, with stability under perturbations measured by multiparameter interleaving distance (Bubenik et al., 2021).

2. Theorematic Foundations and Classical Results

2.1 Persistence of NHIMs in Bounded Geometry

The main result establishes that for $(Q,g)$ of (at least) $(k+1)$-order bounded geometry ($k\geq 2$), any complete $C^{k,\alpha}$-NHIM $M$ with empty unstable bundle persists under $C^{k,\alpha}$-small perturbations of the vector field. More precisely, any $v' \in C^{k,\alpha}_{bu}(Q)$ with $\|v'-v\|_{C^1}$ sufficiently small admits a unique invariant submanifold $M'$:

• $M'$ is $C^{k,\alpha}$ and diffeomorphic to $M$,
• The $C^{k-1}$-distance $\operatorname{dist}_{C^{k-1}}(M, M')$ can be made as small as desired (Eldering, 2012). The persistence holds for time- or parameter-dependent vector fields, yielding $C^{k,\alpha}$-families of invariant manifolds.

2.2 Classical Results—Compactness and Uniqueness

For compact invariant submanifolds, the Hirsch–Pugh–Shub (HPS) theorem asserts that a normally hyperbolic, compact, boundaryless invariant submanifold $N$ persists under $C^1$-small perturbations of the diffeomorphism $f$:

• For every $g$ $C^1$-close to $f$, there exists a unique $C^1$-submanifold $N_g$, diffeomorphic to $N$, with $g(N_g) = N_g$.
• $N_g$ depends smoothly on $g$ and is uniformly locally maximal (Berger et al., 2011).

Noncompact and bounded-geometry generalizations remove compactness assumptions via uniformity conditions (on injectivity radius, curvature), extending the persistence theory to noncompact situations (Eldering, 2012).

2.3 Topological Conditions and Generalizations

Persistence may also arise in the absence of spectral gap conditions. Replacing rate conditions with topological covering relations, Capiński–Kubica (Capinski et al., 2018) show that even if a perturbation destroys normal hyperbolicity, existence of a suitable topological covering relation yields an invariant set (possibly non-manifold), generalizing the classical dynamical persistence paradigm.

3. Technical Hypotheses and Analytical Methods

• Bounded Geometry: Requires a uniform lower injectivity radius, and uniform $C^k$ bounds on the Riemann curvature tensor and its derivatives, ensuring uniform existence and control of normal neighborhoods, uniform charts, and transition functions of bounded geometry (Eldering, 2012).
• Perron (Lyapunov–Perron) Method: The core construction for the persistence of NHIMs in noncompact/bounded-geometry settings builds the perturbed manifold as the graph of a small $C^{k,\alpha}$ map $h: X \to Y$ in a product bundle $X \times Y$ using a variation-of-constants contraction in Banach spaces of exponentially decaying curves (Eldering, 2012).
• Graph Transform and Schauder Fixed Point: For compact NHIMs, the graph transform applied to (Lipschitz/smooth) sections in a tubular neighborhood and use of the Schauder fixed point theorem yield existence (existential non-uniqueness in weak hyperbolicity cases) and smoothness (Berger et al., 2011).

4. Persistent Homology and Approximate Manifolds from Data

Persistent manifold structures also emerge in topological data analysis (TDA). For a manifold sampled by a finite set $X$, persistent homology is computed over filtrations $\{K_\varepsilon\}$, and a plateau of Betti numbers matching the true $H_*(M)$ over a scale interval indicates an approximate persistent manifold structure.

• Theoretical guarantees (Hausmann, Niyogi–Smale–Weinberger) assert that for Hausdorff distance $d_H(X, M) < \tau/2$, the homology of $K_\varepsilon$ matches $M$ for $2d_H(X, M) < \varepsilon < \tau - d_H(X, M)$ (Knudson, 2020).
• For manifolds with "bottleneck" or anisotropic sampling, ellipsoid-based filtrations recover true persistent features more robustly and with better interval stability compared to classical Euclidean ball filtrations (Kališnik et al., 2024).
• In practice, persistent triangulations are built for complex manifolds such as Grassmannians by sampling in the appropriate embedding (e.g., projection matrices), using VR or witness complexes, and verifying plateau regions in the barcode corresponding to the theoretically predicted Betti numbers (Knudson, 2020, Chepushtanova et al., 2016).

5. Sheaf-Theoretic and Multiparameter Formulations

Persistent manifold theory extends via categorical and sheaf-theoretic frameworks:

• For a smooth map $f: M \to N$ between compact manifolds, Curry constructs a "sheaf of chain complexes" on $N$, with stalkwise homology being the homology of the fibers, yielding persistent Betti numbers over open sets and stability (interleaving) under perturbations (Patel, 2011).
• MacPherson–Patel's "persistent local systems" associate to any proper constructible map $f: X \to M$ a family of local systems on open sets $U \subset M$ giving lower bounds $P(U)$ on the Betti numbers of fibers, stable under small perturbations of $f$ (MacPherson et al., 2018).
• Multiparameter persistence modules, as developed by Bubenik and Catanzaro, encode the evolution of homology over families—e.g., one-parameter deformations of functions—on manifolds with full stability and Morse-theoretic classification (Bubenik et al., 2021).

6. Applications and Broader Significance

Persistent manifolds unify dynamical systems invariant manifold theory with topological and geometric data analysis. Applications include:

• Singular perturbation and geometric theory on noncompact domains;
• Persistence and stability of invariant manifolds in infinite-dimensional dynamical systems and PDEs;
• Automated manifold triangulation and topological inference from high-dimensional/sampled data (Grassmannians, tori, hyperspectral data, etc.);
• Robust lower bounds on fiber homology for mappings with varying regularity and constructibility.

The introduction of bounded geometry as a replacement for compactness significantly broadens the reach of persistent manifold theory. This enables the extension of the machinery of invariant manifold persistence and persistent homological invariants to previously inaccessible classes of noncompact geometric/dynamical settings and to algorithmic contexts of manifold learning and data analysis (Eldering, 2012, Knudson, 2020, Patel, 2011).

7. Relation to Classical and Modern Frameworks

Persistent manifolds generalize and encompass classical compact manifold persistence results (Fenichel, HPS) (Berger et al., 2011), noncompact NHIM theory (Eldering, 2012), and are compatible with modern categorical and homological approaches—sheaf and cosheaf modules, multiparameter persistence, spectral estimates of persistent Laplacians, and robust computational frameworks for manifold inference (Knudson, 2020, Kališnik et al., 2024, MacPherson et al., 2018, Patel, 2011).

The framework allows for:

• Quantitative, scale-selective reconstruction of manifolds from finite data;
• Dynamical and geometric stability analysis of manifold-type invariants under perturbation;
• Multiscale and multiparametric extensions of persistence concepts in applied, geometric, and algebraic topology.