DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Geodesic triangulation with prescribed boundary arcs

Definition

Let $(M, g)$ be a compact smooth Riemannian surface, possibly with boundary, and let $T = (V, E, F, ϕ)$ be a curvilinear triangulation of $M$ . It is a geodesic triangulation if every edge $e \in E$ not contained in $\partial M$ admits a regular $C^{2}$ embedding $γ_{e} : [0, 1] \to M$ with image $e$ such that $γ_{e} ((0, 1)) \subset Int M$ and $γ_{e} ∣_{(0, 1)}$ is an affinely parametrized geodesic for the Riemannian surface $(Int M, g ∣_{Int M})$ with its Levi–Civita connection. An edge contained in $\partial M$ remains a prescribed regular $C^{2}$ boundary arc and is exempt from the geodesic condition. The endpoint extension need only be the stated $C^{2}$ embedding; no length-minimizing property or existence of a geodesic triangulation is asserted.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, §“The Gauss–Bonnet Theorem,” printed p. 167 (PDF p. 183), lines 6577–6584, defines a smooth triangulation as finitely many curved triangles covering the surface with pairwise intersections along a common vertex or edge. That passage is context for the underlying face-to-face triangulation; it does not specify geodesic edges.

Datar, Lectures on Riemannian Geometry, Lecture 15, §15.1, Definition 15.1.1, printed p. 113 (PDF p. 121), lines 6271–6283, defines a geodesic by vanishing intrinsic acceleration and allows constant geodesics. The edge regularity inherited here excludes the constant case. The boundary-edge exception and the open-interior endpoint convention are explicit local conventions in this definition; no existence theorem is imported.

Facts & Assumptions

Given: A compact smooth Riemannian surface $(M, g)$ , possibly with boundary, and a curvilinear triangulation $T = (V, E, F, ϕ)$ with its finite edge data.

[F1]

Each edge of a curvilinear triangulation is either contained in $\partial M$ or has relative interior in $Int M$ (Curvilinear face-to-face triangulation).

[F2]

Each curvilinear edge is the image of a regular $C^{2}$ embedding of $[0, 1]$ (Curvilinear face-to-face triangulation).

[F3]

In a boundary chart, an interior point has last coordinate greater than zero (Interior and boundary of a manifold with boundary).

[F4]

A boundary chart is a homeomorphism to a relatively open subset of a half-space (Smooth charts, atlases, and structures with boundary).

[F5]

For a supplied Riemannian metric, a Levi–Civita connection is an affine connection compatible with the metric and torsion free (Levi civita connection).

[F6]

A smooth curve on a boundaryless manifold is an affinely parametrized geodesic when its covariant acceleration $D_{t} γ^{'}$ vanishes (Geodesic of an affine connection).

[F7]

A Riemannian metric is smooth and positive definite, with smoothness up to the boundary where boundaries are allowed (Riemannian metric and riemannian manifold).

Verification

technique · Restrict the metric and connection to the open interior, then unpack the two edge cases in the definition

1.1F3F4F7given

Let $U = Int M$ . In a boundary chart, each point of $U$ has last coordinate strictly positive by [F3]. A sufficiently small Euclidean neighborhood of that chart image misses the model boundary, so the chart restricts to an ordinary smooth surface chart on $U$ by [F4]. Thus $U$ is a boundaryless open submanifold, and [F7] restricts to a smooth positive definite metric $g ∣_{U}$ .

2.1F5F6step 1.1

The Levi–Civita connection of $g ∣_{U}$ is the affine connection used in the geodesic equation on $U$ by [F5]. By [F6], saying that $γ_{e} ∣_{(0, 1)}$ is an affinely parametrized geodesic means precisely that its covariant acceleration for this connection vanishes throughout $(0, 1)$ . The condition is imposed on an affine parametrization of the edge image, not on every possible parametrization.

3.1F1F2step 1.1step 2.1given∎

For each edge, [F1] separates the boundary-contained case from the case whose relative interior lies in $U$ . In the second case the open parameter interval maps into $U$ ; [F2] supplies the regular $C^{2}$ extension to any included endpoints, which may lie on $\partial M$ . The boundary-contained case retains its supplied regular arc and has no geodesic requirement. If $M$ or $E$ is empty these universal conditions are vacuous; regularity excludes constant edge maps, and this finite unpacking selects no element from an arbitrary nonempty family.

Depends on

Used by

Finite geodesic triangulation of a compact Riemannian surface Theorem

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)