[Paper Review] Combinatorial Ricci curvature on cell-complex and Gauss-Bonnnet Theorem

Kazuyoshi Watanabe|arXiv (Cornell University)|Mar 24, 2017

Geometric Analysis and Curvature Flows1 references4 citations

TL;DR

This paper introduces a novel combinatorial Ricci curvature on cell-complexes using the Bochner-Weitzenb"ock formula applied to combinatorial differential forms, defined via an $L^2$ inner product and Laplacian on chain complexes. It establishes Gauss-Bonnet-type theorems for graphs and 2-complexes decomposing closed surfaces, showing that the sum of vertex and face Gauss curvatures equals $4\chi(M)$ for 2-complexes and $2\chi(G)$ for graphs.

ABSTRACT

In this paper we present the Ricci curvature on cell-complexes and show the Gauss-Bonnnet type theorem on graphs and 2-complex that decomposes closed surface. The defferential forms on a cell complex is defined as linear maps on chain complex, and Laplacian operates this defferential forms. Then we construct the Bochner-Weitzenböck formula and define the Ricci curvature. This curvature is determined by the combinatorial calculation. We show also the properties of combinatorial vector fields on a cell complex

Motivation & Objective

• To define a new combinatorial Ricci curvature on cell-complexes using differential forms and the Bochner-Weitzenb"ock formula.
• To establish a Gauss-Bonnet-type theorem for finite graphs and 2-dimensional cell-complexes that decompose closed surfaces.
• To define scalar and Gauss curvatures at vertices and faces via trace of the Ricci curvature form under constant weights.
• To generalize smooth Riemannian curvature concepts to discrete cell-complexes using discrete differential forms and weighted chain complexes.

Proposed method

• Define combinatorial differential forms as local linear maps on the cellular chain complex of a regular cell complex.
• Introduce an $L^2$ inner product on combinatorial differential forms using positive weights on cells.
• Define the Laplacian on combinatorial differential forms via the adjoint of the differential operator under the $L^2$ inner product.
• Construct the Ricci curvature using the combinatorial Bochner-Weitzenb"ock formula, involving parallel vector fields and 0- and 2-neighbor vectors.
• Define the Ricci curvature at a cell as a quadratic form involving weights and components of the 1-form on adjacent cells.
• Derive expressions for Gauss curvature at vertices and faces as normalized traces of the Ricci curvature tensor under constant weights.

Experimental results

Research questions

• RQ1 Can a combinatorial Ricci curvature be defined on cell-complexes using discrete differential forms and the Bochner-Weitzenb"ock formula?
• RQ2 Does a Gauss-Bonnet-type theorem hold for finite graphs and 2-complexes that decompose closed surfaces under this curvature definition?
• RQ3 How does the Ricci curvature at a vertex or face depend on local cell structure and weights?
• RQ4 Is the Gauss curvature at a vertex or face independent of the choice of unit vector under constant weights?
• RQ5 Can scalar curvature be defined as the trace of the Ricci curvature, and does it yield a topological invariant under summation?

Key findings

• For a finite simple graph, the sum of Gauss curvatures over all vertices equals $2\chi(G)$, where $\chi(G)$ is the Euler characteristic.
• For a 2-dimensional quasiconvex cell complex decomposing a closed surface, the sum of vertex and face Gauss curvatures equals $4\chi(M)$.
• Under constant weights, the Gauss curvature at a vertex $v$ is $g_v = 4 - \operatorname{deg}(v)$, and at a face $f$ is $g_f = 4 - \operatorname{deg}(f)$.
• The Ricci curvature at an edge $e > v$ is given by $\operatorname{Ric}(\omega)(e>v) = (2 - \#\{0\text{-neighbor vectors of } \tau > \sigma\})(\omega^\tau_\sigma)^2$.
• The scalar curvature at a vertex $v$ is $S(v) = \operatorname{deg}(v)(4 - \operatorname{deg}(v))$, and at a face $f$ is $S(f) = \operatorname{deg}(f)(4 - \operatorname{deg}(f))$.
• When normalized, the scalar curvature yields the Gauss curvature: $g_v = S(v)/\operatorname{deg}(v) = 4 - \operatorname{deg}(v)$ and $g_f = S(f)/\operatorname{deg}(f) = 4 - \operatorname{deg}(f)$.