Intrinsic Space Formulation

Updated 23 November 2025

Intrinsic space formulation is a framework that models systems using only intrinsic variables like metrics and connections, avoiding extraneous coordinate dependencies.
It systematically eliminates unphysical redundancies by quotienting out global symmetries such as translations, rotations, and scalings to yield a fully relational space.
The approach unifies concepts in differential geometry, machine learning, and theoretical physics, offering robust tools for analyzing dynamics and intrinsic geometric structures.

An intrinsic space formulation refers to the construction, analysis, and application of models where the essential degrees of freedom, geometry, or physical laws are specified not in terms of embedding in an ambient space, nor by extrinsic coordinates, but by structures—such as metrics, connections, or dynamical laws—defined entirely in terms of the intrinsic variables appropriate to the system. This perspective underlies regions of contemporary mathematics and physics, ranging from shape space in relational mechanics, to metric learning in manifold-based machine learning, to the coordinate-free formulation of elasticity, integral geometry, and canonical gravity. The intrinsic space approach systematically eliminates unphysical redundancies, privileging a fully relational, often dimensionless and coordinate-independent, characterization of the phenomena under investigation.

1. Core Mathematical Structures of Intrinsic Spaces

The formulation of intrinsic space begins with identifying the natural configuration or phase space $Q$ , typically as a smooth manifold or more general metric/geodesic space. The subsequent construction involves quotienting $Q$ by a group $G$ encoding the symmetries deemed unobservable (translations, rotations, dilations, or gauge transformations), thereby producing the intrinsic space $S=Q/G$ . For $N$ points in $\mathbb{R}^d$ , the similarity group $\mathrm{Sim}(d)$ acts as $x^I \mapsto R x^I + a$ , $x^I \mapsto \lambda x^I$ , with $R \in SO(d)$ , $a \in \mathbb{R}^{d}$ , and $\lambda>0$ ; thus, shape space is $S = Q/\mathrm{Sim}(d)$ and describes the set of all configurations modulo global position, orientation, and scale. The induced geometry on $S$ derives from a degenerate metric on $Q$ (frequently the mass-weighted Euclidean metric), inherited via a Riemannian submersion, so that $g_{ab}(q)$ on $S$ is unique up to isometry and encodes all observable (relational) structure (Vassallo, 13 May 2025).

Beyond classical configuration spaces, intrinsic space can be realized in machine learning via the geometry of latent and data manifolds, where the pullback metric $f^*g_X$ on a latent manifold $Y$ is used to measure the distortion introduced by a nonlinear map $f:Y\rightarrow X$ relative to the intrinsic metric $g_Y$ of $Y$ (Sun, 2018). In elasticity, the intrinsic space of deformations is the set of Riemannian metrics on the body manifold, modulo isometries, making the true deformation a curve in ${\rm Met}(B)$ rather than in an ambient embedding space (Rashad et al., 2023).

2. Dynamics and Geometry on Intrinsic Spaces

Elimination of extrinsic degrees of freedom yields dynamics governed purely by intrinsic geometric data. In the Barbour–Bertotti relational mechanics, dynamics are encoded via the Jacobi action

$S_{\rm Jacobi}[q] = \int d\lambda\, \sqrt{\,2[E-V(q)]\, g_{ab}(q)\,\dot q^a\,\dot q^b\,}$

where $q^a$ are (local) coordinates on $S$ and $g_{ab}$ is the intrinsic metric. This action describes unparameterized geodesics on $S$ , with the arc-length form $ds = \sqrt{g_{ab}dq^a\,dq^b}$ , physically eliminating any absolute time or length (Vassallo, 13 May 2025). Observable quantities become dimensionless and scale-invariant by foundational construction (e.g., interparticle distances are replaced by ratios to the moment of inertia).

In the context of machine learning, the intrinsic discrepancy $D_{\alpha,f}(y)$ , comparing the pullback metric $f^*g_X$ to $g_Y$ via $\alpha$ -divergence between corresponding neighbourhood measures, quantifies local geometric distortion imposed by $f$ . The isometry case ( $f^*g_X = g_Y$ ) yields vanishing divergence, and the approach is invariant under reparameterization of both spaces, rendering it fully intrinsic (Sun, 2018).

For geometric analysis, the intrinsic space can be determined for minimal discs in metric spaces by endowing the domain with a pseudo-metric given by the infimal length of images of curves, yielding a compact geodesic metric space $Z$ that is intrinsically determined by the area-minimizing map, independent of the possibly singular or highly non-Euclidean nature of the ambient space (Lytchak et al., 2016).

3. Rigidity, Uniqueness, and Recovery of Extrinsic Structure

Intrinsic space formulations are intimately related to rigidity and recovery theorems. The classical Theorema Egregium states that Gaussian curvature is intrinsic; Weyl extended this to higher-dimensional even symmetric functions of principal curvatures. Recent advances show that, for $C^2$ hypersurfaces $M^n \subset N^{n+1}$ in a space form of curvature $\bar K$ , under mild conditions (such as non-vanishing odd symmetric functions $\sigma_{2k+1}(A)$ of the normalized second fundamental form $A$ ), the entire shape operator $A$ is a Riemannian invariant, determined by explicit universal polynomials in the intrinsic Riemann tensor (Guan et al., 13 Jul 2025). Thus, all geometric invariants and functionals previously viewed as extrinsic (e.g., power integrals of principal curvatures) become determined by the induced intrinsic metric on $M$ .

For rolling manifolds, the configuration space is the bundle of isometric identifications between tangent spaces, and the no-slip/no-twist constraints define an intrinsic rank- $n$ distribution on this bundle. Such nonholonomic geometric control problems are fully formulated, analyzed, and proved to be controllable (or not) without recourse to any ambient space (1008.18561210.3140).

4. Information Geometry, Learning, and Algorithmic Representations

Intrinsic geometry in data science leverages these principles to provide regularization and interpretability for complex models. By minimizing local geometric divergence between the induced and intrinsic metrics (e.g., via $D_{\alpha,f}$ ), deep networks and manifold-learning algorithms are encouraged to act as local isometries, promoting embeddings and autoencoders that encode true geometric information. The relations between local divergence penalties and established methods (SNE, t-SNE, Elastic Embeddings, Jacobian regularization in autoencoders/VAEs, GP-LVMs) are precise, and the language of intrinsic space unifies these perspectives (Sun, 2018 Rakotosaona et al., 2020).

In combinatorial and computational geometry, intrinsic spaces can be constructed as inverse limits of finite graphs with assigned metrics, yielding compact geodesic metric spaces with well-defined intrinsic curvature, geodesics, and limiting geometric structure, all arising from intrinsic combinatorial data rather than smooth manifolds (Ambroszkiewicz, 2019).

5. Conceptual, Philosophical, and Physical Significance

The ontological status of intrinsic spaces is an area of ongoing debate, especially in relational theories of physics. The shape space $S = Q/\mathrm{Sim}(d)$ is essential for a relational, scale-free description of dynamics, and mathematically encodes all possible configurations. However, only one curve in shape space corresponds to the realized history; the paper "Shape space as a conceptual space" (Vassallo, 13 May 2025) argues that $S$ is best interpreted as a conceptual—rather than physical—space, analogous to cognitive conceptual spaces in theory formation and perception. This avoids modal over-commitment and unmotivated reification.

In quantum physics and geometry, the "intrinsic space" may be a parameter space in which geometric response invariants—such as the many-body Berry curvature or positional-shift tensor relevant to Hall conductivities—are defined entirely in terms of derivatives of the ground-state wavefunction with respect to flux or field parameters, without reference to an underlying Bloch band or translational symmetry (Resta, 22 Oct 2025).

In canonical gravity, intrinsic coordinate reference frames built from scalar invariants (e.g., the four Weyl scalars) yield a fully background-independent, coordinate-free phase space, resolving the "problem of time" and defining observables internal to the gravitational field (Watson et al., 17 Apr 2024). Similarly, internal-space formulations in teleparallel gravity and alternative geometric approaches to elasticity and Finsler geometry demonstrate that all physical content can be specified intrinsic to the relevant fields or bundles (Tomonari, 7 Oct 2024 Rashad et al., 2023 Soleiman et al., 2023).

6. Broader Applications and Open Problems

Intrinsic space formulations have found deep applications across geometry (integral geometry, tube formulas, kinematic formulas for invariant valuations), geometric analysis (minimal surfaces, uniformization, isoperimetric inequalities), mechanics (elimination of gauge and coordinate redundancies), information geometry, and data science (invariant regularization, manifold learning, geometric deep learning). Open problems include the extension of intrinsic integral geometry to more general manifolds, the classification of combinatorially constructed intrinsic spaces, the full algebraic structure of invariants in quaternionic and octonionic space forms, and the geometric and physical ramifications of nonmetricity and torsion in intrinsic gravity models (1204.06042410.04848Ambroszkiewicz, 2019).

The intrinsic space paradigm continues to drive advances in theoretical physics, differential geometry, and applied mathematics, providing both rigorous foundational understanding and a unified formalism for manipulating and reasoning about systems where redundancy-elimination and invariance principles are paramount.