Finite Distance Gauss–Bonnet Construction

• Finite Distance Gauss–Bonnet construction is a set of frameworks that generalize the classical theorem to finite domains, mesh-based and sub-Riemannian settings.
• Discrete methods assign combinatorial curvatures and verify Euler characteristic identities on triangular lattices and other mesh geometries.
• Analytical extensions include applications to gravitational lensing and sub-Riemannian manifolds, enabling precise computation of finite-range curvature effects.

A finite distance Gauss–Bonnet construction refers to any of several frameworks that generalize or discretize the classical Gauss–Bonnet theorem to finite domains, mesh-based geometries, sub-Riemannian settings, or general relativity with spatially bounded domains. Unlike the classical case, which concerns global integrals over closed manifolds, finite distance constructions address the computation of topological invariants, curvatures, or observable quantities over finite regions or at finite domain boundaries, often requiring nuanced definitions of curvature and careful treatment of boundary contributions.

1. Discrete Gauss–Bonnet Construction in Triangular Lattices

In the discrete combinatorial setting, the finite distance Gauss–Bonnet theorem is realized through curvature assignments on subgraphs of the flat triangular tessellation $X$ as developed by Knill (Knill, 2010). The graph $X$ is formed with vertices at $$\{k(1,0) + \frac{l}{2}(1,\sqrt{3})\,|\,k,l\in\Bbb Z\}$$ and edges joining vertices at Euclidean distance 1. Domains %%%%3%%%% are finite induced subgraphs meeting specific smoothness and dimensionality conditions: interior vertices have full cyclic 6-neighborhoods; the boundary forms a 1-dimensional cyclic graph.

For a vertex $p$ in $G$, spheres %%%%6%%%% are defined as the set of vertices at graph-theoretic distance $r$ from $p$. The arc length $|S^G_r(p)|$ denotes the number of edges in the induced subgraph on $S^G_r(p)$. This construction enables the definition of two curvature functions:

• Puiseux second-order boundary curvature: $K(p) = 2|S_1(p)| - |S_2(p)|$, assigned to boundary vertices.
• First-order combinatorial curvature:

$$K_1(p) = \begin{cases} 6 - |S_1(p)|, & p\ \text{interior} \ 3 - |S_1(p)|, & p \in \partial G \end{cases}$$

The discrete Gauss–Bonnet identities are: $$\sum_{p\in \partial G} K(p) = 12\chi(G), \quad \sum_{p\in G} K_1(p) = 6\chi(G)$$ where $\chi(G) = v - e + f$ is the combinatorial Euler characteristic. For compact 2-dimensional graphs locally isomorphic to $X$, the identity generalizes to $\sum_{p\in M} K(p) = 60\chi(M)$ (Knill, 2010).

2. Analytical and Geometric Frameworks for Ends of Surfaces

For harmonic immersions of punctured compact Riemann surfaces into Euclidean space, the finite distance Gauss–Bonnet formula incorporates explicit boundary contributions at each end. Let $S$ be a compact Riemann surface with punctures $p_1,\ldots, p_n$. For an immersion $X:S' \to \mathbb{R}^d$, locally expressed as the real part of an integral of meromorphic forms, each end $p_i$ yields a contribution $\Theta_i = 2\pi m_i$, with $m_i = \max n_{i,k} -1$ the maximum pole order minus one.

The Gauss–Bonnet formula becomes: $$\int_{S'} K\,dA + \sum_{i=1}^n \Theta_i = 2\pi\chi(S')$$ This quantized end contribution ensures finite total curvature even when the conformal structure of ends is highly nontrivial (Connor et al., 2013).

3. Finite Distance Formulations in Sub-Riemannian and Riemannian Manifolds

The passage from Riemannian to sub-Riemannian Gauss–Bonnet-type identities necessitates a finite distance approximation and careful scaling limits. In three-dimensional sub-Riemannian contact manifolds, a Riemannian approximation $\langle\cdot,\cdot\rangle_L$ is introduced, based on rescaling the Reeb direction. For a surface $S \subset M$ without characteristic points, Gaussian curvature $K_L$ and geodesic curvature $k_{g,L}$ are defined via the adapted metric and remain well-defined in the $L \to \infty$ limit: $$\int_S K_{sub}\,dA_{sub} + \int_{\partial S} k_{g,sub}\,ds_{sub} = 2\pi\chi(S)$$ Here, the subscripted integrals denote limits of the corresponding Riemannian measures and curvatures. Uniform convergence and control of the connection forms are crucial in justifying this sub-Riemannian analogue, and scaling of the Riemannian Gauss–Bonnet equation ensures the correct topological invariant is retained (Veloso, 2020).

4. Application in Gravitational Lensing: Finite Distance Deflection

The Gauss–Bonnet theorem can be leveraged to compute physically observable quantities in general relativity, such as the deflection angle of a light ray in a static, spherically symmetric spacetime at finite source and receiver distances. Using the optical metric $\gamma_{ij}$ associated to the spacetime and considering a domain $D$ whose boundary includes the light trajectory and suitable arcs, the finite distance bending angle $\alpha$ satisfies: $$\alpha = -\iint_D K\,dS = \Psi_R - \Psi_S + \phi_{RS}$$ where $\Psi_R,\Psi_S$ are the angles of the trajectory at the receiver and source, and $\phi_{RS}$ is the coordinate angular separation. This construction allows non-asymptotic corrections in specific spacetimes, such as Schwarzschild–de Sitter or Weyl conformal gravity solutions, to be computed precisely using surface integrals of Gaussian curvature and boundary terms (Ishihara et al., 2016).

5. Boundary Condition Sensitivity and Domain Regularity

Each of the above constructions requires domain regularity or boundary smoothness conditions to ensure that curvature assignments, sphere arc-lengths, and local moves (such as "pruning" or "etching" in the discrete case) behave analogously to the smooth continuum case. In discrete settings, 2-dimensionality is enforced via the structure of the unit spheres $S_1^G(p)$; in continuum and sub-Riemannian frameworks, avoidance of characteristic points or singular directions is necessary for the integrals and limiting procedures to be well-defined (Knill, 2010, Veloso, 2020).

6. Generalizations and Extensions

Finite distance Gauss–Bonnet constructions have been extended in several directions:

• For compact 2-manifolds triangulated in analogy with flat lattices, higher multiples appear in curvature–Euler characteristic identities (e.g., $60\chi(M)$ in the second-order discrete formula).
• Additional discrete curvatures, such as $K_2(p) = 12 - |S_2(p)|$, yield further quantized invariants.
• Analytical approaches to harmonic ends permit a unified treatment of singular or non-conformal geometries, yielding explicit quantized contributions per end.
• In gravitational settings, such constructions systematically extract finite-range deflection effects that are otherwise lost in asymptotic treatments, accommodating cosmological or non-flat backgrounds (Knill, 2010, Connor et al., 2013, Ishihara et al., 2016, Veloso, 2020).

These frameworks collectively facilitate the computation and interpretation of curvature, topology, and related invariants in settings where standard global theories are not directly applicable, while maintaining rigorous correspondence with their classical counterparts.