Pull-Back Metrics in Geometry & Data

• Pull-back metrics are defined by smoothly transferring a target manifold’s metric to a source manifold, establishing inherited notions of lengths and angles.
• They are central to complex geometry applications, such as computing Bergman metrics via the pull-back of the Fubini–Study metric and analyzing curvature properties.
• In topological data analysis, pull-back metrics leverage Jacobian eigen-decomposition to measure sensitivity in persistent homology, aiding parameter tuning and performance assessment.

A pull-back metric is a fundamental construction in differential and complex geometry, used to endow a source manifold with a Riemannian or Kähler structure inherited from a target manifold through a smooth map. Formally, given a smooth map $\Phi: M \to N$ between manifolds equipped with metrics, the pull-back metric $\Phi^*g_N$ on $M$ is defined so that the inner product of tangent vectors at each point is measured by mapping them to $N$ and applying the metric there. This concept underlies diverse areas, including Kähler geometry, the study of Bergman metrics, and topological data analysis via persistent homology encodings.

1. Definition and Differential-Geometric Foundations

Let $(M, g_M)$ and $(N, g_N)$ be smooth (typically Riemannian or Kähler) manifolds, and $\Phi: M \to N$ a smooth map. The pull-back metric $g = \Phi^*g_N$ is a symmetric positive semidefinite $(0,2)$-tensor on $M$, given at each $x \in M$ by

$$g_x(u, v) = g_N|_{\Phi(x)}( d\Phi_x(u), d\Phi_x(v) ),\quad u, v \in T_xM.$$

In local coordinates $\{x^i\}$ on $M$ and $\{y^\alpha\}$ on $N$ with Jacobian $J^\alpha_i = \partial y^\alpha/\partial x^i$, the pull-back metric has components

$$g_{ij}(x) = \sum_{\alpha,\beta} J_i^\alpha(x)\,g_{N,\alpha\beta}(\Phi(x))\,J_j^\beta(x).$$

The induced metric equips $M$ with a notion of lengths and angles derived from $N$ via $\Phi$. When $\Phi$ is an immersion, this metric is nondegenerate, and one often studies its curvature properties (Huang et al., 2023, Liang et al., 2023).

2. Bergman Metrics as Pull-Backs of the Fubini–Study Metric

For a connected complex manifold $M$ of dimension $n$, consider its Bergman space $A^2(M)$ of square-integrable holomorphic $(n,0)$-forms (or $L^2$-holomorphic functions on a domain in $\mathbb{C}^n$). Under the hypotheses that $A^2(M)$ is nontrivial, base-point free, and separates holomorphic directions, the Bergman–Bochner map is defined as

$$\Phi: M \longrightarrow \mathbb{C}P^\infty, \quad \Phi(p) = [\varphi_0(p):\varphi_1(p):\cdots]$$

for any orthonormal basis $\{\varphi_j\}$ of $A^2(M)$. The Fubini–Study (FS) metric on $\mathbb{C}P^\infty$ is given in homogeneous coordinates $[Z_0:Z_1:\cdots]$ by

$$\omega_{\mathrm{FS}} = i\,\partial\overline{\partial}\,\log\left(\sum_{j=0}^\infty|Z_j|^2\right).$$

The pull-back of $\omega_{\mathrm{FS}}$ by $\Phi$ recovers the Bergman metric: $$\omega_B = \Phi^*(\omega_{\mathrm{FS}}) = i\,\partial\overline{\partial}\,\log K(p, \overline{p}),$$ where $K(p, \overline{p}) = \sum_j |\varphi_j(p)|^2$ is the Bergman kernel (Huang et al., 2023).

3. Characterizations and Curvature Properties

The curvature properties of pull-back metrics arising from canonical maps like the Bergman–Bochner map are intricately linked to global analytic features of the underlying domain or manifold. For Stein manifolds with appropriate Bergman spaces and Bergman metric $\omega_B$:

• $\omega_B$ has positive constant holomorphic sectional curvature if and only if $A^2(M)$ is finite-dimensional and $M$ is biholomorphic to a domain in some complex projective space $\mathbb{C}P^r$; then $\Phi$ is a holomorphic isometric embedding.
• $\omega_B$ has negative constant holomorphic sectional curvature if and only if $M$ is biholomorphic to the unit ball $\mathbb{B}^n \subset \mathbb{C}^n$ minus a closed pluripolar set.
• Zero curvature cannot occur for Stein manifolds unless the Bergman space has an orthonormal basis of pure monomials, which is impossible for bounded domains (Huang et al., 2023).

This provides a classification of domains whose Bergman metrics exhibit constant curvature, showing equivalence between analytic/geometric properties and the structure of the pull-back metric.

4. Pull-Back Geometry in Topological Data Analysis

In topological data analysis, persistent homology (PH) encodings act as maps $\Phi$ from a data manifold $M$ to a representation space $N$ (e.g., space of persistence diagrams, Hilbert space of persistence images). The metric properties induced on $M$ via the pull-back metric reflect the sensitivity and expressiveness of the PH encoding.

Given a PH vectorization $\Phi(X)$ and the Euclidean metric $g_N$ on the feature space, the pull-back metric on the data manifold has local Gram matrix $G = J^T J$, where $J$ is the Jacobian of $\Phi$. The spectrum and eigenvectors of $G$ quantify directions of maximal and minimal change in the PH representation; large eigenvalues correspond to informative perturbations, while small eigenvalues signal invariance or noise tolerance.

Computationally, for $u \in T_xM$,

$\|u\|_g = \sqrt{u^T G u}$

provides the pull-back norm, and eigen-decomposition of $G$ produces orthonormal directions of maximal sensitivity. This geometric analysis enables direct assessment of how input perturbations influence topological descriptors, guiding encoding parameter selection without additional supervised models (Liang et al., 2023).

5. Algorithmic and Structural Aspects

Pull-back metric computation in the persistent homology context involves:

1. Computing $J = d\Phi_x$ via automatic differentiation.
2. Forming $G = J^T J$.
3. Computing eigenvalues and eigenvectors of $G$.
4. Using spectral properties to assess sensitivity to perturbations and alignment with features of interest.

For hyperparameter tuning, one varies encoding parameters (e.g., Gaussian width in persistence images), recalculates mean pull-back norms or dominant eigenvalues, and selects settings maximizing desired sensitivity or minimizing projection to noise directions. Practical studies show strong correlation (Pearson's correlation > 0.8 in brain artery datasets) between maximal pull-back norm and logistic regression accuracy on downstream tasks, evidencing the interpretive power of pull-back geometry (Liang et al., 2023).

6. Examples and Notable Phenomena

Concrete manifestations of pull-back metrics include:

• Domains in $\mathbb{C}^2$ with Bergman metrics of constant holomorphic sectional curvature $+2$, constructed using explicit defining functions and showing $\{1, z, w\}$ spans $A^2(D(a))$, so $\Phi$ lands in $\mathbb{C}P^2$ and inherits the FS metric (Huang et al., 2023).
• Hartogs–Reinhardt domains in $\mathbb{C}^{n+1}$ with Bergman metrics that are flat along totally geodesic $\mathbb{C}^n$ subspaces, demonstrating the phenomenon of locally flat but globally non-flat pull-back metrics.
• Persistent homology encodings where the directions of maximal pull-back norm (approximated by the largest eigenvectors of $G$) correspond to features most strongly influencing the topological summary, enabling principled sensitivity analyses (Liang et al., 2023).

7. Conjectures, Structural Rigidity, and Future Directions

Conjecture 3.4 posits that for Stein manifolds whose Bergman space is base-point free and separates directions, the Bergman metric cannot have positive constant holomorphic sectional curvature. Known complete Stein–Bergman metrics of constant holomorphic sectional curvature are always non-positive. The existence of unbounded domains with $>0$ curvature (not Stein) suggests a rigidity in the Stein/pseudoconvex setting: no genuinely Stein domain in $\mathbb{C}^n$ can possess a Bergman metric isometric to projective space. A stronger conjecture (3.5) asserts the impossibility of any such manifold having holomorphic sectional curvature bounded below by a positive constant, implying a sharp curvature sign dichotomy for Stein–Bergman metrics (Huang et al., 2023).

In computational topology, the pull-back metric framework is emerging as an intrinsic, model-free criterion to evaluate and interpret persistent homology encodings, with evidence that spectral properties of the induced metric reliably predict downstream task performance (Liang et al., 2023).