Submetric Image of a Carnot Group

• Submetric images are sub-Riemannian manifolds derived via submetries from Carnot groups, ensuring metric balls are exactly preserved.
• They enable the transfer of Taylor polynomial theory and regularity criteria from Carnot groups to their quotient spaces through explicit analytic and geometric mappings.
• These structures underpin innovative representations of signature spaces for rectifiable paths and connect noncommutative analysis with geometric measure theory.

A submetric image of a Carnot group is a sub-Riemannian manifold obtained via a submetry from a Carnot group, where the metric descends to the quotient in a manner that exactly preserves metric balls. This construction generalizes the canonical quotient process for nilpotent Lie groups and has intrinsic significance for the study of sub-Riemannian geometry, noncommutative analysis, and geometric measure theory. The concept of submetric images underlies the structure of signature spaces for rectifiable paths and allows for the transfer of classical analytic and geometric results—such as Taylor polynomials and regularity criteria—to broader noncommutative contexts.

1. Carnot Groups and Submetry

A Carnot group $G$ is a connected, simply–connected nilpotent Lie group whose Lie algebra $\mathfrak{g}$ admits an $s$–step stratification,

$$\mathfrak{g} = V_1 \oplus V_2 \oplus \cdots \oplus V_s, \quad [V_1, V_j] = V_{j+1} \ (1 \leq j < s), \quad V_{s+1} = \{0\}.$$

Key structures include exponential coordinates, homogeneous dilations $\delta_\lambda$, and the left-invariant horizontal distribution $H_gG = (L_g)_*(V_1)$, with a corresponding Carnot–Carathéodory distance,

$$d_G(g_1, g_2) = \inf_{\gamma(0)=g_1,\,\gamma(1)=g_2} \int_0^1 \|\dot\gamma(t)\|_{\gamma(t)}\,dt,$$

where $\gamma$ is horizontal. A map $\pi: X \rightarrow Y$ between metric spaces is called a submetry if it carries balls onto balls of equal radius,

$$\pi\bigl(B_X(\pi^{-1}(y), r)\bigr) = B_Y(y, r), \quad \forall r \geq 0,\, y \in Y,$$

and is $1$-Lipschitz. Submetries preserve key geometric features while allowing for the projection of analytic structures (Donne et al., 2019, Ottazzi, 13 Dec 2025).

2. Construction of Submetric Images via Quotients

Given a Carnot group $G$ and a closed subgroup $H \subset G$ invariant under dilations, the right–coset space

$M = H \backslash G$

inherits a quotient metric defined by

$d_M(Hg_1, Hg_2) = \inf_{h \in H} d_G(g_1, h g_2),$

rendering the projection $\pi: G \to M$ a submetry. The induced horizontal bundle $H_pM = \pi_*(H_gG)$ and left-invariant sub-Riemannian metric on $M$ are canonically determined via lifts in $G$. The resulting manifold $(M, d_M)$ is termed a submetric image of $G$ (Ottazzi, 13 Dec 2025). The submetry property ensures metric balls descend faithfully, preserving the sub-Riemannian geometry.

3. Inverse Limits and the Space of Signatures

For applications to path signatures, one considers the sequence of free nilpotent Carnot groups $\{G_k\}_{k \geq 1}$ with fixed rank $n$ and increasing step $k$. The bonding maps $\pi_{k+1, k}: G_{k+1} \to G_k$ “forget” the top layer and are explicit group homomorphisms, each a $1$–Lipschitz submetry. The inverse limit

$$G_\infty = \lim_{\longleftarrow}\{G_k, \pi_{k+1,k}\}$$

is equipped with the majoration metric

$$d_\infty\bigl((g_k), (h_k)\bigr) = \sup_{k \geq 1} d_k(g_k, h_k) < \infty,$$

yielding a uniquely geodesic, infinitely branching metric tree structure. Hambly–Lyons's uniqueness theorem identifies $G_\infty$ with the space of (un-truncated) signatures of rectifiable paths in $\mathbb{R}^n$: every geodesic in $G_\infty$ projects to a rectifiable path, and conversely, any path in $\mathbb{R}^n$ can be approximated by such projections. The complexity and structure of geodesics in these groups reflect the iterated integration captured at each step (Donne et al., 2019).

4. Analytic and Taylor Structures on Submetric Images

The submetric image construction allows the transfer of Taylor polynomial theory and regularity criteria. On Carnot groups, Taylor polynomials of homogeneous degree $k$ at $g_0$ satisfy

$$\left(\tilde X^I \tilde P_k(F,g_0)\right)(g_0) = \left(\tilde X^I F\right)(g_0), \quad \forall\, I \text{ with } d(I) \leq k,$$

where $\tilde X^I$ are iterated left-invariant horizontal vector fields. These results extend to quotient spaces $M = H \backslash G$ by push-forward of vector fields and functions,

$$P_k(f, p) = \tilde P_k(f \circ \Pi, \Phi^{-1}(p)) \circ L_{\Phi^{-1}(p)}^{-1}$$

where $\Phi^{-1}$ is a local section. Explicit mean-value and Taylor remainder estimates hold, with constants depending only on $(G, H)$. Higher-order expansions (Peano-type) and regularity criteria for analyticity follow by standard arguments (Ottazzi, 13 Dec 2025).

5. Real-Analyticity and Harmonicity on Submetric Images

Functions $f \in C^\infty(M)$ satisfying a factorial growth condition on the horizontal derivatives,

$$\sup_{d_M(p,q)<\rho} |X^I f(q)| \leq K^{d(I)}\, d(I)!$$

are real-analytic on $M$. This is proved by lifting to $G$, using the submetry property and Bonfiglioli’s analytic theorem for Carnot groups. Furthermore, Taylor polynomials $P_n(f,p)$ on $M$ are $L$–harmonic for any left-invariant, homogeneous differential operator $L$ annihilating $f$. This is established by explicit lifting of operators and polynomials to $G$ and then pushing forward harmonicity to the quotient (Ottazzi, 13 Dec 2025).

6. Geometric and Functional Implications

The submetric image paradigm provides a robust framework for representing and analyzing the geometry and analysis of sub-Riemannian quotients. Metric properties—including geodesicity, ball structure, and path-lifting—are preserved under submetry. This suggests a general mechanism by which analytic and geometric results for Carnot groups can be faithfully inherited by their submetric images, including but not limited to Taylor expansions, regularity, and harmonicity. A plausible implication is that the study of submetric images enables a transfer principle for regularity and approximation theory in sub-Riemannian geometry, with direct relevance for analysis on left-quotients, control theory, and the geometry of signature spaces.

7. Connections and Further Developments

The identification of the space of signatures with the metric tree $G_\infty$ establishes deep ties between Carnot group theory and the analysis of rough paths and iterated integrals. Submetric images underpin explicit geometric realizations of analytic phenomena in quotient spaces, with consequences for approximation schemes in metric geometry and the theory of horizontal regularity. Future investigations may clarify the role of submetries in the mapping theory of metric measure spaces, the fine structure of sub-Riemannian function spaces, and the universality of geodesic spaces arising as inverse limits of Carnot groups (Donne et al., 2019, Ottazzi, 13 Dec 2025).