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Gauss-Bonnet for a geodesic polygon

Statement

Assume the axiom of choice. Let $(M, g, J)$ be an oriented Riemannian surface and let $D \subseteq M$ be a positively oriented compact regular disk region whose boundary is the cyclic concatenation of finitely many regular $C^{2}$ geodesic segments of the interior metric, with ordinary corners at the vertices and no other corners. Then, with the signed exterior angles $α_{1}, \dots, α_{m}$ of the positively oriented boundary,

$\int_{D} K d A + \sum_{j = 1}^{m} α_{j} = 2 π .$

Facts & Assumptions

Given: Full AC through the local disk Gauss–Bonnet supplier (The Axiom of Choice); A positively oriented compact regular disk region whose boundary consists of finitely many geodesic segments meeting at ordinary corners.

[F1]

For every positively oriented compact regular disk region with finitely many ordinary corners, $\int_{D} K d A + \int_{\partial D} k_{g} d s + \sum_{j} α_{j} = 2 π$ (Local Gauss-Bonnet for an arbitrary disk region).

[F2]

Every side of the boundary admits a regular $C^{2}$ geodesic parametrization whose interior is affinely parametrized (Geodesic of an affine connection).

[F3]

The signed geodesic curvature is the scalar with covariant acceleration $A_{γ} = k_{g} J T$ , so it vanishes wherever the covariant acceleration vanishes (Signed geodesic curvature).

Proof

technique · insert the vanishing geodesic curvature of the sides into the arbitrary-disk formula

1.1F2F3given

Each boundary side is a geodesic segment, so on its interior the covariant acceleration of its unit-speed parametrization vanishes by [F2], and therefore its signed geodesic curvature vanishes identically by [F3].

2.1F1step 1.1algebra∎

The boundary integral over the finitely many sides vanishes by step 1.1, so [F1] applied to $D$ reads $\int_{D} K d A + \sum_{j} α_{j} = 2 π$ .

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, Theorem 9.3, printed pp. 165-167, contains the formula with the boundary curvature term; setting $k_{g} = 0$ along geodesic sides gives the stated corollary. Datar, Lectures on Riemannian Geometry, Lecture 2, Theorem 2.0.1, printed pp. 10-13, gives the same formula. The vanishing of $k_{g}$ along geodesics is the library definition Signed geodesic curvature applied to Geodesic of an affine connection.

Depends on

Used by

Corner terms are required even in the plane Counterexample
Area excess of a spherical geodesic triangle Example

Dependency tree · two levels

19 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)