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Local Gauss-Bonnet for an arbitrary disk region

Statement

Assume the axiom of choice. Let $(M, g, J)$ be an oriented Riemannian surface and let $D \subseteq M$ be a compact regular oriented disk region with finitely many ordinary corners, of the kind fixed by Regular oriented surface regions with corners. Then

$\int_{D} K d A + \int_{\partial D} k_{g} d s + \sum_{j = 1}^{m} α_{j} = 2 π,$

where $k_{g}$ is the signed geodesic curvature of the positively oriented boundary and $α_{1}, \dots, α_{m}$ are its signed exterior angles. No global orthonormal frame on a neighbourhood of $D$ is required, and the full-choice assumption is inherited from the frameable decomposition and the frameable disk formula.

Facts & Assumptions

Given: An oriented Riemannian surface and a compact regular oriented disk region with finitely many ordinary corners.

[F1]

$D$ admits a finite face-to-face subdivision into regular disk pieces $D_{1}, \dots, D_{F}$ such that each piece lies in an oriented coordinate chart of $M$ and carries a smooth positive orthonormal frame, with only ordinary corners. Every new edge is piecewise smooth, and the relative interior of each new non-boundary edge lies in $Int D$ ; its endpoints may lie on $\partial D$ (Finite frameable decomposition of a regular disk region).

[F2]

For a positively oriented compact regular disk region with ordinary corners carrying a smooth positive orthonormal frame on a neighbourhood, $\int_{D} K d A + \int_{\partial D} k_{g} d s + \sum_{j} α_{j} = 2 π$ (Local Gauss-Bonnet for a frameable disk region).

[F3]

At a positively oriented boundary corner with interior sector angle $β \in (0, 2 π)$ , the signed exterior angle is $α = π - β$ (Signed exterior angle at an ordinary corner).

[F4]

Reversing the parameter of a regular $C^{2}$ unit-speed curve changes the sign of its signed geodesic curvature at every point (Signs of geodesic curvature under reversals).

[F5]

The finite triangular data of [F1] form a regular CW structure on the closed disk $D$ , so $V - E + F = \sum_{j} (- 1)^{j} rank H_{j} (D; Z) = 1$ because a disk is contractible (Euler–Poincare formula for finite CW complexes, Finite frameable decomposition of a regular disk region).

Proof

technique · apply the frameable formula to each piece of the finite frameable decomposition, cancel internal edges, count vertices with the planar disk identity, and read off the boundary terms

1.1F1given

Apply [F1] to obtain the finitely many regular disk pieces $D_{i}$ , each contained in an oriented chart of $M$ and carrying a smooth positive orthonormal frame on a neighbourhood; the pieces are face-to-face, all corners are ordinary, and the subarcs of $\partial D$ appear as boundary sides of the pieces.

2.1F1F2step 1.1

Every piece $D_{i}$ is a compact regular oriented disk region with ordinary corners carrying a smooth positive orthonormal frame on a neighbourhood, so [F2] applies to it: $\int_{D_{i}} K d A + \int_{\partial D_{i}} k_{g} d s + \sum_{v} α_{v}^{(i)} = 2 π$ , the sum being over the piece's corners.

3.1step 2.1algebra

Summing step 2.1 over the finitely many pieces and using additivity of the area integral over the face-to-face decomposition gives $2 π F = \int_{D} K d A + \sum_{i} \int_{\partial D_{i}} k_{g} d s + \sum_{i} \sum_{v} α_{v}^{(i)}$ .

4.1F1F4step 3.1

Each internal edge of the decomposition is incident with exactly two pieces and is traversed by them in opposite directions, because both pieces inherit the orientation of $D$ ; by [F4] the two signed curvature integrals over that edge cancel. Each subarc of $\partial D$ is incident with exactly one piece and carries the positive boundary orientation, so the surviving edge integral is $\int_{\partial D} k_{g} d s$ .

5.1F1F3step 4.1algebra

At every decomposition vertex $v$ let $m_{v}$ be the number of piece corners at $v$ and $β_{v}^{(i)} \in (0, 2 π)$ the interior angle of the corresponding piece; by [F3] the piece contributes $α_{v}^{(i)} = π - β_{v}^{(i)}$ . Summing over all piece corners gives $\sum_{i} \sum_{v} α_{v}^{(i)} = π \sum_{v} m_{v} - (2 π V_{i n t} + π V_{b d}^{s u b} + \sum_{j} β_{j}^{o r i g})$ , where $V_{i n t}$ counts interior vertices, $V_{b d}^{s u b}$ counts boundary vertices subdividing smooth boundary arcs, and $β_{j}^{o r i g}$ are the interior angles at the original corners of $D$ . In a face-to-face decomposition into disk cells $\sum_{v} m_{v} = 2 E_{i n t} + E_{b d}$ and $E_{b d} = V_{b d} = V_{b d}^{s u b} + m$ for the number $m$ of original corners, so $\sum_{i} \sum_{v} α_{v}^{(i)} = \sum_{j} α_{j}^{o r i g} + 2 π (E_{i n t} - V_{i n t})$ .

6.1F1F5step 5.1

The finite CW count of [F5] applies to the decomposition: $V - E + F = 1$ . Since boundary edges and boundary vertices occur in equal numbers, this reads $V_{i n t} - E_{i n t} + F = 1$ , equivalently $F - E_{i n t} + V_{i n t} = 1$ .

7.1step 3.1step 4.1step 5.1step 6.1algebra∎

Substituting steps 4.1, 5.1 and 6.1 into step 3.1 gives $2 π F = \int_{D} K d A + \int_{\partial D} k_{g} d s + \sum_{j} α_{j}^{o r i g} + 2 π E_{i n t} - 2 π V_{i n t}$ , hence $\int_{D} K d A + \int_{\partial D} k_{g} d s + \sum_{j} α_{j} = 2 π (F - E_{i n t} + V_{i n t}) = 2 π$ .

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, Theorem 9.3 together with the reduction preceding Theorem 9.7, printed pp. 162-169, proves the local formula first for a region contained in a chart and then passes to general regions; Datar, Lectures on Riemannian Geometry, Lecture 2, Theorem 2.0.1, printed pp. 10-13, gives the same computation. The reduction is carried out here through the library's finite frameable decomposition Finite frameable decomposition of a regular disk region, the internal-edge cancellation, the corner bookkeeping and the disk Euler count of Finite planar graph disk cuts and Euler count.

Depends on

Used by

Gauss-Bonnet for a geodesic polygon Corollary

Dependency tree · two levels

22 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)