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Flat closed oriented surfaces have Euler characteristic zero

Statement

Assume the axiom of choice. Let $(M, g)$ be a nonempty closed oriented Riemannian surface whose Gaussian curvature vanishes identically, $K \equiv 0$ on $M$ . Then $χ (M) = 0$ , where $χ$ is the Euler characteristic of Topological well-definedness of the surface Euler characteristic.

Facts & Assumptions

Given: A nonempty closed oriented Riemannian surface $(M, g)$ with $K \equiv 0$ .

[A1]

full AC is assumed; it is inherited exactly through the global Gauss-Bonnet theorem and is used nowhere else (The Axiom of Choice).

[F1]

For every closed oriented Riemannian surface, $\int_{M} K d A = 2 π χ (M)$ with $d A$ the area form of the orientation and $χ$ the invariant of the well-definedness theorem (Global Gauss-Bonnet for closed oriented surfaces).

[F2]

The area form $d A$ of a specified orientation is the Riemannian volume form, and the integral of a compactly supported top form is defined chartwise; in particular the zero top form has integral $0$ (Riemannian volume form on an oriented manifold, Integral of a compactly supported top form).

Proof

technique · insert the identically vanishing curvature into the global identity and divide by $2\pi$

1.1F2given

Since $K \equiv 0$ , the top form $K d A$ is the zero two-form at every point of $M$ ; by the definition of the compactly supported top-form integral, $\int_{M} K d A = 0$ .

2.1F1step 1.1algebra

By [F1], $\int_{M} K d A = 2 π χ (M)$ . Substituting step 1.1 gives $2 π χ (M) = 0$ , and since $2 π \neq 0$ it follows that $χ (M) = 0$ .

3.1A1step 2.1∎

No new choice is made: the only use of full AC is the inherited one through the global theorem, and the computation is a division by the nonzero constant $2 π$ in the real numbers.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, Theorem 9.7, printed pp. 167-172, proves $\int_{M} K d A = 2 π χ (M)$ 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 identity. The flat case $K \equiv 0$ is an immediate specialization, performed here with the library's top-form integral of Integral of a compactly supported top form and the Euler characteristic of Topological well-definedness of the surface Euler characteristic.

Depends on

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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)