Relative Coordinate Descriptor (RCD)

• Relative Coordinate Descriptor is a framework that defines a coordinate system on noncompact metric measure spaces using Busemann functions and transport rays.
• It employs measure disintegration to separate spaces into radial and transversal components, facilitating the study of volume growth, splitting, and rigidity phenomena.
• This approach generalizes classical geometric results to nonsmooth settings, providing tools for optimal transport, spectral analysis, and structural theorems.

The Relative Coordinate Descriptor (RCD) is a framework arising from the paper of noncompact metric measure spaces with synthetic lower Ricci curvature bounds and dimension upper bounds, formalized by the RCD(0, N) condition. Within this context, the RCD description provides a coordinate system tailored to the intrinsic geometry of these spaces, facilitating analysis and geometric measure-theoretic results by decomposing the space into "radial" and "transversal" components via Busemann functions and their associated transport rays. This approach generalizes classical constructions on smooth manifolds to the broader setting of nonsmooth metric measure spaces, allowing for the paper of splitting phenomena, volume growth, and rigidity results.

1. RCD(0, N) Spaces and Linear Volume Growth

The RCD(0, N) condition is a "Riemannian" curvature-dimension condition imposed on a metric measure space $(X, d, m)$, combining the requirements of being infinitesimally Hilbertian (so that the Sobolev space $W^{1,2}(X,d,m)$ is Hilbert) and satisfying a synthetic Ricci curvature lower bound ($K=0$) and dimension upper bound $N$ (Huang, 2016). Noncompact RCD(0, N) spaces necessarily exhibit at least linear volume growth. The minimal linear volume growth case is characterized by

$$\limsup_{R \to \infty} \frac{m(B_p(R))}{R} = V_0 < +\infty,$$

where $B_p(R)$ denotes the metric ball of radius $R$ centered at $p \in X$.

This volume growth constraint precludes excess volume at infinity and anchors the subsequent geometric analysis.

2. Busemann Functions and Foliation

Given a geodesic ray $\gamma:[0,\infty)\to X$, the associated Busemann function $b:X\to\mathbb{R}$ is defined by the limit

$b(x) = \lim_{t \to \infty} (t - d(x,\gamma(t))),$

where $d$ is the metric on $X$. The function $b$ is 1-Lipschitz, and its level sets $b^{-1}(r)$ (the "horizons") are crucial to relative coordinate construction.

A central result states that under minimal volume growth conditions, the diameters of these level sets grow at most linearly: $$\limsup_{R \to \infty} \frac{\operatorname{diam}(b^{-1}(R))}{R} \leq C_0,$$ with $C_0 \leq 2$. Furthermore, each level set $b^{-1}(r)$ is compact.

These horizons yield a foliation of the space, providing a "radial coordinate" system indexed by the Busemann function.

3. Disintegration, Transport Rays, and the Relative Coordinate Framework

A key aspect of the RCD relative coordinate description involves disintegrating the measure $m$ along the level sets of $b$ and characterizing points via the unique transport rays associated with these level sets. Specifically, for almost every point $x$ with $b(x) > 0$, there is a unique geodesic (transport ray) along which $b$ increases linearly.

This leads to the mapping

$g: S \times \mathbb{R}_+ \to X,$

where $S$ is a Borel cross-section (often a subset $Z = b^{-1}(r_0)$), and each $x$ (with $b(x) > 0$) is represented by coordinates $(z, t)$

$$x = g(z, t), \quad \text{with } z = P_i(x) \in Z,\, t = b(x).$$

Projecting onto $S$ provides the "transversal" coordinate, while $t$ acts as the radial coordinate. The flow maps

$F_t(x) = g(P_i(x), b(x) + t)$

are locally isometric and measure-preserving, establishing a product structure at infinity: $$X \cong Z \times (r_0, \infty), \quad Z = b^{-1}(r_0),$$ with induced transverse metric.

This decomposition enables the analysis of metric measure spaces through a system resembling generalized polar coordinates, the "relative coordinate description" (Editor's term).

4. Strongly Minimal Volume Growth and Splitting Theorems

With "strongly minimal volume growth," the measure-preserving flow preserves Dirichlet energy, and the Busemann function is both harmonic and satisfies

$$\Delta b = 0, \quad \text{and} \quad \text{Hess}(b) = 0, \quad \text{for } m\text{-a.e. on } b^{-1}((0, \infty)).$$

These rigidity conditions facilitate a splitting theorem: at infinity, $X$ either splits as a product $Z \times \mathbb{R}_+$ (with $Z$ itself an RCD(0, N–1) space) or, in the trivial case, is isometric to a half-line $[r_0,\infty)$.

Consequently, the relative coordinate framework is not merely local; under strong volume growth constraints, it extends to global structural results that subsume classical splitting theorems.

5. Connections to Classical Geometry and Analysis

The RCD relative coordinate framework parallels and generalizes several classical theorems, such as the Cheeger–Gromoll splitting theorem and Sormani’s work on Busemann functions for smooth manifolds with nonnegative Ricci curvature. Essential features available in the smooth category—product structure at infinity, rigidity under volume growth, foliations by level sets—are shown to persist in the synthetic and nonsmooth regime under the RCD condition. These methods substantially broaden the analytic and geometric toolkit for the paper of singular spaces and the convergence behavior of manifolds.

Applications include:

Classical Context	RCD Relative Coordinates	Consequence
Ricci nonnegativity (smooth)	RCD(0, N) curvature-dimension condition	Foliation, Splitting
Busemann function analysis	1-Lipschitz, harmonic Busemann in RCD	Measure disintegration
Cheeger–Gromoll splitting	Product at infinity, rigidity in RCD	Structure theorem

6. Analytical Implications and Optimal Transport

Disintegration of the measure along transport rays is closely connected to optimal transport theory. The well-defined ray maps and associated coordinate systems form the backbone of modern analysis on metric measure spaces, facilitating the paper of isoperimetric inequalities, spectral behavior, and rigidity phenomena. The relative coordinate descriptor provides a systematic way to adapt arguments from smooth Riemannian geometry to singular spaces, capturing both local and global geometric information in the noncompact, nonsmooth setting.

7. Summary and Structural Significance

In summary, the Relative Coordinate Descriptor organizes noncompact RCD(0, N) spaces into a product-like structure dictated by Busemann functions and their level sets, enabling measure-theoretic and geometric analysis via splitting, rigidity, and optimal transport techniques (Huang, 2016). Such a framework extends the reach of classical geometric and analytic theorems to the broad, nonsmooth category, making it an indispensable tool in the modern understanding of spaces with synthetic Ricci curvature bounds.