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Global Gauss-Bonnet for closed oriented surfaces

Statement

Assume the axiom of choice through the parent triangulation and metric-extension suppliers. Let $M$ be a closed oriented Riemannian surface, that is, a compact oriented smooth surface with empty boundary carrying a Riemannian metric $g$ . Then $\int_{M} K d A = 2 π χ (M),$ where $K$ is the Gaussian curvature, $d A$ the area form of the orientation, and $χ (M)$ the Euler characteristic of Topological well-definedness of the surface Euler characteristic. For the empty surface $M = \emptyset$ the identity is the true statement $0 = 0$ , since $χ (\emptyset) = 0$ .

Facts & Assumptions

Given: A closed oriented Riemannian surface, possibly empty and possibly disconnected, with its metric and orientation.

[A1]

Full AC is inherited exactly through the parent theorem's curvilinear triangulation supplier and is used nowhere else (The Axiom of Choice).

[F1]

A closed oriented Riemannian surface is a compact oriented Riemannian surface with smooth (empty) boundary; when it is viewed as a compact regular region in itself, the regular-region convention admits the empty boundary, and the parent theorem gives $\int_{M} K d A + \int_{\partial M} k_{g} d s + \sum_{j} α_{j} = 2 π χ (M)$ with both boundary terms omitted when $\partial M = \emptyset$ (Gauss-Bonnet for compact oriented surface regions with boundary and corners, Regular oriented surface regions with corners).

[F2]

A curvilinear triangulation of the empty surface has $V = E = F = \emptyset$ , so its count is $χ (\emptyset) = 0 - 0 + 0 = 0$ , and for a nonempty compact smooth surface the invariant $χ (M)$ is the common value $V - E + F$ of the finite curvilinear triangulations (Topological well-definedness of the surface Euler characteristic).

Proof

technique · apply the boundary-and-corners theorem with empty boundary and split off the empty-surface case

1.1A1F1F2given

If $M \neq \emptyset$ , view $M$ as a compact oriented Riemannian surface presented in presentation (a) of the parent theorem with $Σ = M$ and empty boundary, or equivalently in presentation (b) with empty smooth boundary; by [F1] the theorem applies and both the boundary integral and the corner sum are omitted, giving $\int_{M} K d A = 2 π χ (M)$ , with $χ (M)$ the common count of [F2].

1.2F2givenalgebra

If $M = \emptyset$ , then every finite curvilinear triangulation has no vertices, edges or faces by [F2], so $χ (\emptyset) = 0$ ; the integral of a function over the empty surface is $0$ and the identity reads $0 = 2 π \cdot 0$ , which is true.

2.1A1step 1.1step 1.2∎

In both cases the asserted identity holds, and the full-choice assumption was inherited unchanged from the parent theorem through its triangulation supplier; no new choice, orientation cover or classification statement is used.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, Theorem 9.7, printed pp. 167-172, proves the global formula $\int_{M} K d A = 2 π χ (M)$ for a compact oriented surface without boundary; Datar, Lectures on Riemannian Geometry, Lecture 2, Theorem 2.2.4, printed pp. 14-15, gives the same statement. The reduction to the boundary-and-corners theorem with empty boundary is performed here using the library conventions of Regular oriented surface regions with corners, and the Euler characteristic is the invariant of Topological well-definedness of the surface Euler characteristic; the empty-surface case is handled from the triangulation-indexed definition rather than by convention.

Depends on

Used by

Flat closed oriented surfaces have Euler characteristic zero Corollary
Metric independence of total Gaussian curvature Corollary
Positive curvature forces positive Euler characteristic Corollary
Flat torus and zero Euler characteristic Example
Total curvature of a round sphere Example
Gauss-Bonnet alone does not classify surfaces False statement
Scope of classical surface Gauss-Bonnet Remark

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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)