Geodesics for Discrete manifolds

TL;DR

This work builds a discrete differential-geometric framework for finite simplicial complexes by defining a geodesic flow as a permutation on a discrete frame bundle, enabling geodesic sheets and a robust notion of sectional curvature. Sectional curvature is derived from dual cells within geodesic sheets, with Gauss-Bonnet-type relations connecting form curvatures to Euler characteristics and Ishida–Higuchi curvature in low dimensions. It introduces partition curvature as a combinatorial criterion for positivity and presents a Poincaré–Hopf interpretation via index-expectation, unifying energy redistribution and curvature in a discrete setting. The results suggest discrete sphere-type phenomena, offer tools for discrete billiards and ergodic questions, and point to further links with classical Cartan-type geometry in a finite framework.

Abstract

The geodesic flow on a finite discrete q-manifold with or without boundary is defined as as a permutation of its ordered q-simplices. This allows to define geodesic sheets and a notion of sectional curvature.

Figures (5)

• Figure 1: The map $T$ maps a totally ordered facet $x$ into its adjacent facet. To the picture to the left: here $(1,2,3,4)$ is an ordered simplex in a 3-manifold. There exists exactly one dual simplex $(2,3,4,5)$. The direction of the flow depends on the orientation. The orientation defines a base point $1$ and an ordered triple of edges $\{ (12),(13),(14) \}$ that can be thought of as a basis with base $1$. The geodesic flow transports that basis to the next simplex $\{ (23),(24),(25) \}$.
• Figure 2: When reaching boundary parts, the map does a self rotation on the simplex until a continuation is possible. It produces a discrete billiard map. The map can be defined for all pure simplicial complexes for which the stable sphere $S^+(x)=U(x) \setminus \{x\}$ has $1$ or $2$ elements only for all $(q-1)$-simplices.
• Figure 3: We see a 2-manifold $G$, the greatrhombicuboctahedron, an example of a 2-sphere. Its dual $\hat{G}$ is the disdyakisdodecahedron. The Eberhard curvatures $1-d(v)/6$ of $G$ are $1/3,0,-1/3$. The bones are the q-2=0 simplices in $G$ which are the vertices. There are three types of geodesic sheets defining sectional curvature$(2-4)/6 + (2/3)(2/8+2/6)=1/18$ appearing 12 times, $(2-6)/6 + (2/3)(3/4+3/8) = 1/12$ appearing 8 times and $(2-8)/6 + (2/3)(4/4+4/6) = 1/9$ appearing 6 times. Unlike the first order Eberhard curvature, these second order curvatures are all positive. They are also the partition curvatures$\frac{(2-m)}{6}+\sum_{i=1}^m \frac{1}{p_i}$ of the partitions $p=(8,8,6,6), p=(8,8,8,4,4,4)$ and $p=(6,6,6,6,4,4,4,4)$ which encode the possible vertex degrees of a vertex and its neighbors. The partitions belong to the unit sphere degree 28, 36, 40. We remark below that if the unit sphere degree is less or equal than 31, then curvature is automatically positive.
• Figure 4: We see a 3-ball with 2-sphere boundary given as the soft Barycentric refinement of the icosahedron. We chose a random bone $x$ (here an edge as $q=3$ and $q-2=1$). We first see the dual bone, a set of m=5 simplices hinging on the bone. We use then the geodesic flow to extend this to a discrete two dimensional geodesic sheet which is the dual graph of a 2-ball. The petal numbers are $p=(6,6,6,6,6)$ in this case. The geodesic sheet is made of 20 tetrahedra. The sectional curvature is the partition curvature of $p$ which is $K(x)=\frac{2-5}{6} + (\frac{3}{2})(\sum_{j=1}^5 \frac{1}{6}) = 1/18$.
• Figure 5: A small $3$-manifold $G=\mathbb{R}\mathbb{P}^3$ with $f$-vector $f=(11,51,80,40)$ and Euler characteristic $\chi(G)=11-51+80-40=0$. Its refinement $G_1$ has $f$-vector $f(G_1)=(182,1142,1920,960)$. We see one of the possible geodesic sheets of a central bone of order $5$ surrounded by bones of order $p=(5,5,5,4,4)$, a partition of $23$. The sectional curvature is $(2-5)/6 + (2/3)\sum_{i=1}^5 \frac{1}{p_i} = 7/30$. All sectional curvatures of $G$ are either $1/5$ or $7/30$. For the Barycentric refinement the sectional curvatures are all in $\{1/9, 1/6, 2/9, 1/3\}$. Both $G$ and $G_1$ are positive curvature manifolds.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5: 31 theorem
• proof