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Complete connected Riemannian manifolds are proper length spaces

Statement

Assume ${A C}_{ω}$ . Let $(M, g)$ be a nonempty, connected, boundaryless Riemannian manifold whose Riemannian distance $d_{g}$ is complete. Then $(M, d_{g})$ is a proper length space in the following precise sense:

every closed bounded subset of $(M, d_{g})$ is compact;
$d_{g} (x, y)$ is the infimum of the Riemannian lengths of piecewise- $C^{1}$ curves from $x$ to $y$ ; and
for every $x, y \in M$ this infimum is attained by a minimizing geodesic.

Facts & Assumptions

Given: The manifold and completeness hypothesis in the statement.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the assumed ${A C}_{ω}$ , and Boundaryless convention for geodesic flow and Hopf–Rinow fixes the boundaryless convention.

[F1]

Riemannian distance on a connected manifold defines $d_{g} (x, y)$ as the infimum of the lengths of piecewise- $C^{1}$ curves from $x$ to $y$ .

[F2]

Under [A1], Hopf–Rinow theorem says that on a nonempty connected boundaryless Riemannian manifold, completeness of $(M, d_{g})$ implies both compactness of every closed bounded subset and existence, for each $x, y$ , of a geodesic of length $d_{g} (x, y)$ .

Proof

technique · direct

1.1

The completeness hypothesis is condition 1 of [F2]. Its equivalent condition 5 proves clause 1, and its final assertion supplies for each $x, y$ the minimizing geodesic in clause 3.

2.1

Clause 2 is exactly the definition in [F1]. Combining it with clause 3 shows not only that $d_{g}$ is the induced length metric, but that its defining infimum is achieved.

F1step 1.1

3.1

Nonemptiness, connectedness and absence of boundary are exactly the hypotheses required by [F2]; none is discarded. In dimension zero, $M$ is a singleton, its only closed bounded subsets are empty or singleton, and the constant geodesic realizes distance zero. Dimension one needs no change. At $x = y$ the minimizing geodesic is constant. Empty subsets are covered by clause 1, and there are no radius or endpoint divisions in the proof. The corollary is one-way, so no converse is asserted. Its sole choice assumption is [A1], used through [F2]; reading the defining infimum in [F1] adds no choice.

A1F1F2step 1.1step 2.1∎

Source locator

Datar, Theorem 19.2.1 and proof, pp.141--144: metric completeness implies the properness and minimizing-geodesic conclusions read off here.

Depends on

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Sources

Ved Datar, Lectures on Riemannian Geometry, Theorem 19.2.1 and proof, pp.141--144 (standard reference, not scraped)