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A local isometry from a complete connected manifold has geodesically complete target image

Statement

Assume ${A C}_{ω}$ . Let $F : (M, g) \to (N, h)$ be a local Riemannian isometry between boundaryless Riemannian manifolds. Suppose $M$ is connected and complete for its Riemannian distance, and put $O = F (M)$ .

Then $O$ is open in $N$ and, with the restricted Riemannian metric, is geodesically complete. More strongly, for every $q \in O$ and $w \in T_{q} N$ , the maximal $N$ -geodesic with initial data $(q, w)$ is defined for all real time and its entire image lies in $O$ .

No injectivity, surjectivity onto $N$ , or covering-map conclusion is asserted.

Facts & Assumptions

Given: The local isometry and completeness hypotheses 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 isometry and local isometry says that a local Riemannian isometry is a smooth local diffeomorphism satisfying $F^{*} h = g$ ; in particular every $d F_{p}$ is a linear isomorphism and every $p$ has a neighbourhood mapped diffeomorphically onto an open subset of $N$ .

[F2]

Under [A1], Hopf–Rinow theorem makes a nonempty connected boundaryless Riemannian manifold that is complete for its Riemannian distance geodesically complete. Geodesically complete Riemannian manifold says this means that every maximal geodesic has domain $R$ .

[F3]

Under [A1], Existence uniqueness and smooth dependence of geodesics supplies the unique maximal geodesic for each initial tangent vector and identifies every other geodesic with the same initial data as its restriction.

[F4]

Local isometries send geodesics to geodesics says that a local Riemannian isometry sends affinely parametrized geodesics to affinely parametrized geodesics and intertwines their velocities.

Proof

technique · lift initial data

1.1

For $q \in O$ , choose $p \in M$ with $F (p) = q$ . By [F1], some neighbourhood of $p$ maps diffeomorphically onto an open neighbourhood $V$ of $q$ in $N$ , and $V \subseteq F (M) = O$ . Thus every point of $O$ is interior and $O$ is open. The choice of $p$ instantiates one existential statement for one fixed $q$ .

F1given

2.1

If $O = \emptyset$ , it has no initial tangent vectors and is geodesically complete vacuously by [F2]; the stronger assertion is vacuous as well. Suppose $q \in O$ and fix $w \in T_{q} N$ . Choose one $p \in M$ with $F (p) = q$ . Since $O$ is open, $T_{q} O = T_{q} N$ , and [F1] gives the unique vector $v = (d F_{p})^{- 1} w \in T_{p} M .$

F1F2step 1.1

3.1

The point $p$ shows that $M$ is nonempty, so [F2] applies to the assumed metric completeness of $M$ . By [F3], the maximal source geodesic $η : R \to M$ with initial data $(p, v)$ is therefore defined for every real time.

F2F3step 2.1

4.1

Regard $F$ first as a map $M \to O$ with the restricted target metric. It is still a local Riemannian isometry by [F1], so [F4] makes $σ = F \circ η : R \to O$ a geodesic. It has $σ (0) = q$ and $σ^{'} (0) = d F_{p} (v) = w$ . By [F3], the maximal geodesic in $O$ with data $(q, w)$ contains this global geodesic and hence has domain $R$ . Since $(q, w)$ was arbitrary, $O$ is geodesically complete.

F1F3F4step 3.1

5.1

Regard the same $σ$ as an $N$ -valued geodesic. It has the same initial data $(q, w)$ , so uniqueness in [F3] identifies it with the maximal $N$ -geodesic on that geodesic's domain. Because $σ$ itself is defined on all of $R$ , maximality makes the $N$ -geodesic global; its value at every time is $F (η (t)) \in O$ . This proves the stronger assertion.

F3F4step 4.1

6.1

The empty case was settled in step 2.1. In dimension zero, every tangent vector is zero and the relevant geodesics are constant; dimension one is unchanged. The zero vector and both positive and negative infinite-time directions are included because $η$ has domain all of $R$ . The statement is one-way and does not infer a covering map. Assumption [A1] is used through [F2] and [F3]; choosing one preimage after fixing $(q, w)$ and applying one inverse linear map require no family-wide choice.

A1F1F2F3step 1.1step 2.1step 3.1step 4.1step 5.1∎

Source locator

Datar, Theorem 20.1.1 and its geodesic-lifting step, pp.147--149, prove the stronger covering and target-completeness theorem for a complete source local isometry. The present corollary retains only the initial-data lifting and geodesic-complete-image consequences.

Depends on

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Sources

Ved Datar, Lectures on Riemannian Geometry, Theorem 20.1.1 and geodesic-lifting step, pp.147--149 (standard reference, not scraped)