Technical Summary: Recovering Riemannian Geometry from Diffusion

Problem Statement

Traditional geometric analysis typically begins with a pre-existing Riemannian structure (a manifold $M$, a metric $g$, and a measure $\mu$) and studies the associated diffusion processes, such as the heat flow generated by the Laplace–Beltrami operator. This paper reverses that ontological direction. The central problem addressed is whether the full Riemannian geometry of an underlying space can be intrinsically reconstructed solely from the data of a symmetric, strongly local diffusion semigroup $(P_t)_{t \geq 0}$, without assuming any prior metric, coordinate system, or heat kernel representation.

The paper posits that the geometric structure is not an a priori input but an emergent property encoded in the infinitesimal behavior of diffusion. Specifically, it asks if the first- and second-order diffusion calculi are sufficient to recover the metric tensor, the Levi–Civita connection, the curvature tensor, and the reference measure.

Methodology

The reconstruction proceeds through the Bakry–Émery $\Gamma$-calculus, utilizing the intrinsic operators associated with the diffusion semigroup. The methodology relies on the following logical chain:
$$\text{Diffusion Semigroup } (P_t) \longrightarrow \text{Bakry–Émery Calculus } (\Gamma, \Gamma_2) \longrightarrow \text{Riemannian Geometry } (g, \nabla, \mu)$$

1. Foundational Assumptions

The paper assumes a smooth, connected manifold $M$ equipped with a $\sigma$-finite Borel measure $\mu$ of full support. The diffusion is modeled by a conservative, symmetric Markov $C_0$-semigroup $(P_t)$ on $L^2(M, \mu)$ with a self-adjoint generator $L$.
Key axioms include:

• Strong Locality: The associated Dirichlet form $\mathcal{E}$ is strongly local.
• Smoothness: The diffusion admits a smooth first-order calculus. Specifically, the carré du champ operator $\Gamma$ is well-defined on a smooth core $C^\infty(M)$, is symmetric, bilinear, satisfies the Leibniz rule, and depends only on first-order jets.

2. The Reconstruction Steps

The paper constructs the geometric objects in the following order:

• Recovering the Metric ($g$):
  The carré du champ $\Gamma(f, g)$ is defined intrinsically as the limit of the decorrelation of the semigroup:
  $$\Gamma(f, g) := \lim_{t \downarrow 0} \frac{1}{2t} \left( P_t(fg) - (P_t f)(P_t g) \right).$$
  The paper demonstrates that $\Gamma(f, g)(x)$ depends only on the differentials $df(x)$ and $dg(x)$. This allows the definition of a symmetric bilinear form on the cotangent bundle $T^*_x M$. Under a non-degeneracy condition (positive definiteness), this form defines a co-metric $g^{-1}$. Its inverse yields a unique smooth Riemannian metric $g$ such that $\Gamma(f, g) = \langle df, dg \rangle_{g^{-1}}$.

• Recovering the Connection ($\nabla$) and Measure ($\mu$):
  Once $g$ is established, the paper shows that the symmetry of the diffusion semigroup rigidly selects the Levi–Civita connection. The generator $L$ is shown to take the form of a weighted Laplacian:
  $$L = \Delta_g + \langle \nabla \log \rho, \nabla \cdot \rangle_g,$$
  where $\mu = \rho \, \text{vol}_g$. The drift term $\nabla \log \rho$ is uniquely determined by comparing $L$ with the standard Laplace–Beltrami operator $\Delta_g$, thereby recovering the reference measure $\mu$ up to a global multiplicative constant.

• Recovering Curvature:
  The iterated carré du champ $\Gamma_2$ is defined via the generator as $\Gamma_2(f) = \frac{1}{2}L\Gamma(f) - \Gamma(f, Lf)$. The paper utilizes the Bochner identity to show that $\Gamma_2$ encodes the Bakry–Émery Ricci tensor:
  $$\Gamma_2(f) = \|\nabla^2 f\|_{HS, g}^2 + (\text{Ric}_g + \nabla^2 \log \rho)(\nabla f, \nabla f).$$
  By choosing functions with vanishing Hessian at a point, the Ricci tensor (and subsequently the full Riemann curvature tensor via the connection) is recovered pointwise from $\Gamma_2$.

Key Contributions and Results

1. Intrinsic Metric Recovery: The paper proves that a symmetric, strongly local diffusion semigroup satisfying smoothness axioms determines a unique smooth Riemannian metric $g$. This metric is derived purely from the first-order calculus $\Gamma$, without recourse to geodesics or heat kernel asymptotics.
2. Uniqueness of Connection and Measure: The symmetry of the diffusion semigroup uniquely fixes the Levi–Civita connection and the reference measure $\mu$. Any other affine connection or measure would violate the fundamental identities satisfied by $\Gamma$ and the Dirichlet form.
3. Curvature from Second-Order Data: The iterated carré du champ $\Gamma_2$ is shown to encode the weighted Ricci curvature. This establishes that curvature is an intrinsic property of the second-order diffusion behavior.
4. Global Isometry Determination: The paper establishes that the diffusion semigroup determines the global weighted Riemannian manifold $(M, g, \mu)$ uniquely up to Riemannian isometry. If two diffusion systems are "diffusion-equivalent" (conjugate via a measurable bijection), their reconstructed geometries are globally isometric.
5. Information-Theoretic Perspective: The work provides a rigorous framework where geometric structure emerges from the flow of information. The diffusion calculus serves as a complete invariant of the Riemannian isometry class.

Significance and Claims

The paper claims to provide a foundational shift in differential geometry by demonstrating that geometry can be viewed as the structure governing information flow under diffusion.

• No A Priori Geometry: The reconstruction requires no initial metric, coordinates, or spectral data. The metric is an emergent property of the first-order diffusion calculus.
• Rigidity: The results show a high degree of rigidity; the diffusion data uniquely determines the geometric package $(g, \nabla, \text{Ric}, \mu)$.
• Scope: The paper explicitly states that these results apply to smooth manifolds with smooth diffusion calculi. It does not claim to solve the reconstruction problem for non-smooth spaces, discrete structures, or non-commutative analogues, though it suggests these as potential directions for future investigation.
• Relation to Prior Work: The paper builds upon earlier work (cited as [1]) where the smooth manifold structure itself was recovered from entropy. This paper assumes the smooth structure is available and focuses on recovering the Riemannian structure (metric, connection, curvature) from the diffusion semigroup.

In summary, the paper establishes that the "intrinsic behavior of diffusion" contains all the data necessary to reconstruct the full weighted Riemannian geometry of the underlying space, effectively reversing the standard paradigm of deriving diffusion from geometry.