Riemannian Pullback Metric: Theory & Applications

• Riemannian pullback metric is a construction in differential geometry that transfers a metric from a target manifold to a domain via smooth maps.
• It is applied in fields such as geometric data analysis, latent manifold learning, and shape analysis, reducing complex geometries to tractable computations.
• Algorithmic strategies like automatic differentiation and closed-form Jacobians enhance the efficiency, stability, and accuracy of its practical implementations.

The Riemannian pullback metric is a fundamental construction in differential geometry and applied mathematics, enabling the transfer of Riemannian geometric structure from a target space to a domain manifold via a smooth map, typically a diffeomorphism or a more general differentiable mapping. This device is central to contemporary methodologies in geometric data analysis, generative modeling on constrained matrix spaces, latent manifold learning, harmonic analysis, and the generalized theory of shape spaces and symmetric bundles.

1. Definition and General Construction

Given a Riemannian manifold $(N, g_N)$ and a smooth manifold $M$, a smooth map (frequently a global diffeomorphism) $\varphi: M \rightarrow N$ allows the definition of a pullback metric $\varphi^* g_N$ on $M$ by

$$(\varphi^*g_N)_x(v, w) := g_N(\varphi(x))\left(D\varphi(x)[v], D\varphi(x)[w]\right),\quad v, w \in T_x M.$$

In coordinate notation, if $g_N$ at $\varphi(x)$ is represented by a $d\times d$ positive-definite matrix $G_N(\varphi(x))$ and $J_\varphi(x)$ is the Jacobian, then the pullback metric at $x$ is given by

$$g_M(x) = J_\varphi(x)^\top \, G_N(\varphi(x))\, J_\varphi(x).$$

This construction determines how infinitesimal displacements in $M$ are “measured” using the geometry of $N$ after mapping through $\varphi$. The setting generalizes to stochastic mappings (e.g., in latent variable models), in which case the metric may be random or equipped with an uncertainty structure (Augenstein et al., 28 Oct 2024, Rozo et al., 7 Mar 2025).

2. Classical and Modern Examples

2.1 Matrix Manifold Generative Models

For manifold-valued data such as symmetric positive definite (SPD) or correlation matrices, $M$ may be $\mathrm{Sym}^+_n$ or the set of correlation matrices, and practical diffeomorphisms $\varphi$ of interest include the matrix logarithm $\varphi(\Sigma) = \log(\Sigma)$ or the normalized Cholesky decomposition $\varphi(\Sigma) = \operatorname{vecl}(D^{-1/2} L)$, where $L$ is the Cholesky factor and $D$ its block-diagonal (Collas et al., 20 May 2025).

For the log-Euclidean metric on SPD matrices, the pullback metric formula is

$$g^\varphi_\Sigma(\Xi, \eta) = \operatorname{Tr}\left(D\log(\Sigma)[\Xi] \cdot D\log(\Sigma)[\eta]\right)$$

with

$$D\log(\Sigma)[\Xi] = \int_0^\infty (\Sigma + sI)^{-1} \Xi (\Sigma + sI)^{-1} ds.$$

This enables reductions of complex Riemannian structures to Euclidean vector space computations via the mapping $\varphi$ (and its inverse).

2.2 Latent Space Learning and Autoencoding

In latent variable models, such as Gaussian Process Latent Variable Models (GPLVMs), one considers $f: Z \to X$ with $Z$ a low-dimensional “latent” manifold and $X$ a data manifold, then endows $Z$ with a pullback metric $f^* g_X$ using the Jacobian $Df(z)$ (and possibly the native Riemannian metric on $Z$ in case $Z$ is curved, e.g., hyperbolic). In Gaussian process settings, the metric becomes a random object determined by a matrix normal or non-central Wishart distribution and can be computed in closed form for expectations (Augenstein et al., 28 Oct 2024, Rozo et al., 7 Mar 2025).

Score-based approaches use a density model $p(x) \propto \exp(-\psi(\vartheta(x)))$ with $\vartheta$ a diffeomorphism, yielding a composite pullback metric through the gradient map $\varphi = \nabla\psi \circ \vartheta$ and its (regularized) Jacobian (Diepeveen et al., 2 Oct 2024).

2.3 Harmonic Maps and Pullback Geometry

For a smooth map $f: (M, g) \to (N, h)$, the pullback metric $f^* h$ provides the local inner product structure on $M$ induced by $N$. The energy density and its trace, as well as harmonicity criteria, can be expressed directly in terms of $f^* h$. The divergence and trace properties underpin geometric characterizations such as the harmonic identity map criterion (Stepanov et al., 10 Jul 2025).

2.4 Applications to Tangent Bundles and Path Spaces

In the paper of the geometry of curves and their shape analysis, pullback metrics arise naturally in constructions such as the square root velocity function (SRVF) formalism, where the pullback of a (generalized) Sasaki metric from the tangent bundle yields a first-order Sobolev metric with desirable invariance properties (Brigant et al., 2015, Vaisman, 2013).

3. Algorithmic Implementation and Computational Aspects

The pullback metric structure admits efficient algorithmic exploitation whenever the map $\varphi$ is explicit and computationally tractable. Key computational steps include:

• Evaluation of the Jacobian $J_\varphi(x)$, which may be achieved via automatic differentiation, finite differences, or closed-form expressions for structured maps (matrix logarithm, Cholesky, exponential map).
• Norms, inner products, and geodesics in $M$ reducing to their Euclidean (or target-manifold) counterparts via the coordinate representation.
• Sampling, flow-based learning, and vector field parameterization are efficiently executed in the image (e.g., Euclidean) space and then “pulled back” to the manifold as required (Collas et al., 20 May 2025).

Specific functional forms, such as those for log-Euclidean SPD geometry or normalized Cholesky, may require computational subroutines for eigen-decomposition or operations on lower-triangular matrix factorizations, but complexity is mitigated by moving most computation into Euclidean coordinates.

Where the metric is stochastic (e.g., in GPLVMs), one works with expected metric tensors or samples from their posteriors as needed for uncertainty quantification (Augenstein et al., 28 Oct 2024, Rozo et al., 7 Mar 2025).

4. Geodesics, Distances, and Interpolation

The principal geometric effect of the pullback metric is that geodesics and distances in $M$ are mapped to more tractable (often straight-line) computations in the target space:

• For a global diffeomorphism $\varphi$, a geodesic $\Gamma(t)$ in $M$ corresponds to a straight line $(1-t)\varphi(x_0)+t\varphi(x_1)$ in the image space, which can then be pulled back by $\varphi^{-1}$ (Collas et al., 20 May 2025).
• In latent space models and autoencoders, the geodesic between encoded points is computed by minimizing curve energy relative to the pullback metric, often using spline parameterizations or graph-based shortest paths when the metric must be computed numerically (Augenstein et al., 28 Oct 2024, Tennenholtz et al., 2021).
• In probabilistic and stochastic settings, geodesics respect both the base curvature and the uncertainty structure (e.g., variance inflation away from data support), which ensures interpolation passes through or near high-data-density regions. This mechanism avoids the undesirable “shortcut” effects of naive geodesics in negatively curved spaces (Augenstein et al., 28 Oct 2024, Rozo et al., 7 Mar 2025, Diepeveen et al., 2 Oct 2024).

Empirical results consistently show that pullback metrics offer improved geometric fidelity, faithful interpolation within real data regions, and uncertainty control, supporting applications in neural generative modeling, trajectory generation, diffusion models on manifolds, and manifold-aware data augmentation (Collas et al., 20 May 2025, Augenstein et al., 28 Oct 2024, Rozo et al., 7 Mar 2025).

5. Stability, Regularization, and Theoretical Guarantees

The utility of the pullback metric in applications depends critically on regularity properties of the mapping $\varphi$ and properties of the target geometry (Diepeveen, 11 Mar 2024):

• Properness (geodesic completeness), stability (continuous dependence under data or map perturbations), and efficiency (preservation of local distances) are jointly achieved by ensuring $\varphi$ maps the data manifold into a totally geodesic or flat submanifold of the target space and is locally isometric on the data support.
• Local isometry (singular values of $D\varphi|_{T_xX}$ near 1) ensures minimal curvature distortion and robust metric behavior even under small deformations or noise.
• Curvature of the target (e.g., positive for $S^2$, negative for $H^2$) introduces variability in interpolation and barycenter computations: pulling back negative curvature yields stability, whereas positive curvature may amplify instability, as confirmed by empirical examples (Diepeveen, 11 Mar 2024).

Isometry regularization is often incorporated in deep learning constructions of $\varphi$ (e.g., invertible residual networks) to maintain the desired metric fidelity (Diepeveen et al., 2 Oct 2024).

6. Applications Across Domains

The pullback metric underpins a broad set of methodological advances:

• Manifold-aware flow-based generative models for SPD/correlation matrices with all operations lifted to Euclidean coordinates for computational efficiency (Collas et al., 20 May 2025).
• Geometry-aware latent variable models for hierarchical and structured data, ensuring that geodesics, interpolations, and uncertainty estimates follow the true (data) manifold (Augenstein et al., 28 Oct 2024, Rozo et al., 7 Mar 2025).
• Shape analysis and invariant metrics on spaces of immersions or paths, permitting geodesic interpolation for shape and curve analysis (Brigant et al., 2015).
• Structured data analysis on symmetric Riemannian manifolds, ensuring efficiency and stability in non-linear embedding and learning procedures (Diepeveen, 11 Mar 2024).
• Construction of fast, closed-form, and robust alternative metrics for information-theoretic and statistical manifolds through pullback of projective or Finslerian metrics, leading to computational benefits in large-scale tasks (Nielsen, 2023).
• Uncertainty quantification and control in offline reinforcement learning via data-driven Riemannian metrics on latent representations, directly influencing policy robustness and acquisition of data-conservative solutions (Tennenholtz et al., 2021).

7. Relation to Broader Geometric Structures

The pullback metric construction extends beyond Riemannian geometry to Finsler geometry, as in the pullback of projective Hilbert metrics, which yield non-quadratic “Minkowski norms” on tangent spaces and preserve smoothness and computational tractability (Nielsen, 2023). In tangent bundle geometry, the pullback of generalized Sasaki metrics admits connections with torsion and curvature entirely encoded in the base geometry and additional model data, offering flexibility for applications in complex and Kähler geometry (Vaisman, 2013).

The general framework is compatible with the harmonic map theory, providing analytic tools for the characterization of energy-minimizing maps, trace decomposition, and connections to harmonic symmetric tensors and metrics (Stepanov et al., 10 Jul 2025).

In summary, the Riemannian pullback metric serves as a unifying paradigm for inducing, transferring, and exploiting geometric structure on spaces equipped with nonlinear mappings to tractable or computationally favorable domains, with robust theoretical underpinnings and demonstrable algorithmic and empirical benefits across modern geometric data science.