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The cut locus of a point is closed

Statement

Assume exactly the inherited Axiom of Countable Choice ${A C}_{ω}$ , carried by the declared exponential-domain, cut-time and sequential-closure suppliers. Let $(M, g)$ be a complete, connected, boundaryless, finite-dimensional Riemannian manifold, let $p \in M$ , and let $Cut (p) = {\exp_{p} (c_{p} (v) v) : v \in S_{p} M, c_{p} (v) < + \infty}$ be the cut locus of $p$ from Cut point and cut locus of a point. Then $Cut (p)$ is closed in the metric space $(M, d_{g})$ .

In dimension zero $S_{p} M = \emptyset$ , so $Cut (p) = \emptyset$ , which is closed. No compactness of $M$ is assumed; the only compactness used is the sequential compactness of the finite-dimensional unit sphere $S_{p} M$ .

Facts & Assumptions

Given: The complete connected boundaryless finite-dimensional Riemannian manifold $(M, g)$ , the point $p \in M$ , the cut time $c = c_{p} : S_{p} M \to (0, + \infty]$ , the unit sphere $S_{p} M$ and the cut locus $Cut (p)$ .

[A1]

The choice assumption is ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ), inherited through the declared suppliers and spent in this proof exactly at the two points flagged in steps 1.1 and 5.1; no full Axiom of Choice and no dependent choice is used.

[F1]

The cut locus is $Cut (p) = {\exp_{p} (c_{p} (v) v) : v \in S_{p} M, c_{p} (v) < + \infty}$ , and along the unit-speed radial geodesic in direction $v$ one has $γ_{v} (t) = \exp_{p} (t v)$ (Cut point and cut locus of a point).

[F2]

If the cut time $c_{p} (v)$ is finite then $c_{p} (v) \in A_{p} (v) = {t \geq 0 : d_{g} (p, γ (t)) = t}$ , that is $d_{g} (p, γ_{v} (c_{p} (v))) = c_{p} (v)$ (Minimizing along a geodesic is an initial interval property).

[F3]

$d_{g}$ is a finite metric on the connected Riemannian manifold $M$ (Riemannian distance is a metric), so it satisfies (M1) separation, (M2) symmetry and (M3) the triangle inequality (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric); convergence $x_{k} \to x$ in $(M, d_{g})$ means $d_{g} (x_{k}, x) \to 0$ in $R$ (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ).

[F4]

$S_{p} M$ is sequentially compact: every sequence of unit tangent vectors at $p$ has a subsequence converging in the norm metric of $T_{p} M$ to a unit tangent vector at $p$ (Unit spheres in finite-dimensional normed spaces are sequentially compact).

[F5]

The cut time is positive at every unit vector, and it is continuous in the extended sense: whenever $v_{k} \to v$ in $S_{p} M$ , if $c (v) < + \infty$ then $c (v_{k}) \to c (v)$ , while if $c (v) = + \infty$ then for every $M_{0} > 0$ there is $k_{0}$ with $c (v_{k}) > M_{0}$ for all $k \geq k_{0}$ (Cut time is positive and continuous).

[F6]

On the complete manifold $(M, g)$ the Hopf–Rinow equivalent condition 3 holds: for every point of $M$ the fibre exponential domain is all of the tangent space, so $E_{p} = T_{p} M$ (Hopf–Rinow theorem).

[F7]

The exponential map is smooth on its domain, hence continuous; consequently each fibre restriction $\exp_{p}$ is smooth on the open set $E_{p}$ (The exponential domain is open and the exponential map is smooth). The target manifold topology is the $d_{g}$ topology (The riemannian distance topology is the manifold topology), so this continuity also gives convergence in $(M, d_{g})$ .

[F8]

On $T_{p} M$ the Riemannian metric is a positive definite symmetric bilinear form (Riemannian metric and riemannian manifold), the pointwise norm is $∣ v ∣_{g} = \sqrt{g (v, v)}$ (Pointwise norm and angle from a riemannian metric), and the induced inner-product norm satisfies $∣ λ v ∣_{g} = ∣ λ ∣ ∣ v ∣_{g}$ and $∣ u + v ∣_{g} \leq ∣ u ∣_{g} + ∣ v ∣_{g}$ (The induced length is a norm).

[F9]

In a metric space a subset $F$ is closed if and only if it is sequentially closed, that is, if and only if every sequence in $F$ that converges in the ambient space has its limit in $F$ (A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed).

Proof

technique · sequential closedness, read through the metric-space equivalence between closed and sequentially closed sets: a convergent sequence of cut points has cut times equal to the distances to $p$, hence bounded; compactness of the unit sphere produces a limit direction; continuity of the cut time makes the limiting cut time finite; and continuity of the exponential map realises the limit as the corresponding cut point

1.1A1F1F2F3given

A convergent sequence of cut points and its cut times. [A1, F1, F2, F3, given] Let $(q_{k})$ be a sequence in $Cut (p)$ with $q_{k} \to q$ in $(M, d_{g})$ . For every $k$ the set ${v \in S_{p} M : c_{p} (v) < + \infty, \exp_{p} (c_{p} (v) v) = q_{k}}$ is nonempty by [F1], and selecting one $v_{k}$ for each $k \in N$ is exactly the countable choice permitted by [A1]; then $c (v_{k}) < + \infty$ and $q_{k} = \exp_{p} (c (v_{k}) v_{k})$ for every $k$ . By [F2] a finite cut time is attained, $c (v_{k}) \in A_{p} (v_{k})$ , so $c (v_{k}) = d_{g} (p, γ_{v_{k}} (c (v_{k}))) = d_{g} (p, \exp_{p} (c (v_{k}) v_{k})) = d_{g} (p, q_{k})$ , using $γ_{v} (t) = \exp_{p} (t v)$ [F1]. Since $d_{g}$ is a metric [F3], (M2) and (M3) give $∣ d_{g} (p, q_{k}) - d_{g} (p, q) ∣ \leq d_{g} (q_{k}, q)$ for every $k$ , and $d_{g} (q_{k}, q) \to 0$ because $q_{k} \to q$ [F3]; hence $c (v_{k}) \to d_{g} (p, q) = : L$ with $L \in [0, + \infty)$ , so the real sequence $(c (v_{k}))$ is bounded.

2.1F3F4givenstep 1.1

A convergent subsequence of directions. [F3, F4, given, step 1.1] Since $(c (v_{k}))$ is bounded (step 1.1) and $S_{p} M$ is sequentially compact [F4], there is a subsequence $v_{k_{j}} \to v$ with $v \in S_{p} M$ . A subsequence of the convergent sequence $(c (v_{k}))$ has the same limit $L$ [F3, step 1.1], so $c (v_{k_{j}}) \to L$ . Relabelling that subsequence, assume from now on that $v_{k} \to v$ in $S_{p} M$ and $c (v_{k}) \to L$ .

3.1F3F5givenstep 2.1

The limiting cut time is finite. [F3, F5, given, step 2.1] Suppose $c (v) = + \infty$ . By the infinite-value clause of the continuity of the cut time [F5], for every $M_{0} > 0$ there is $k_{0}$ with $c (v_{k}) > M_{0}$ for all $k \geq k_{0}$ ; taking $M_{0} = L + 1$ contradicts $c (v_{k}) \to L$ (step 2.1) [F3]. Hence $c (v) < + \infty$ , and the finite-value clause of [F5] gives $c (v_{k}) \to c (v)$ . The two limits $c (v)$ and $L$ of the same sequence agree: for every $ε > 0$ both $∣ c (v_{k}) - c (v) ∣ < ε$ and $∣ c (v_{k}) - L ∣ < ε$ hold for all large $k$ , whence $∣ c (v) - L ∣ < 2 ε$ , and $ε > 0$ was arbitrary. Therefore $c (v) = L < + \infty$ .

4.1F1F6F7F8givenstep 1.1step 2.1step 3.1

The limit point is the corresponding cut point. [F1, F6, F7, F8, given, step 1.1, step 2.1, step 3.1] Put $w_{k} : = c (v_{k}) v_{k}$ and $w : = c (v) v$ . The vectors $w_{k}$ lie in $E_{p}$ because $\exp_{p} (w_{k}) = q_{k}$ [F1], and by [F6] $E_{p} = T_{p} M$ , so also $w \in E_{p}$ . For the convergence $w_{k} \to w$ , the norm estimates of [F8] give, since $∣ v_{k} ∣_{g} = ∣ v ∣_{g} = 1$ and $c (v_{k}) > 0$ , $∣ w_{k} - w ∣_{g} = ∣ c (v_{k}) v_{k} - c (v_{k}) v + c (v_{k}) v - c (v) v ∣_{g} \leq c (v_{k}) ∣ v_{k} - v ∣_{g} + ∣ c (v_{k}) - c (v) ∣ ∣ v ∣_{g} = c (v_{k}) ∣ v_{k} - v ∣_{g} + ∣ c (v_{k}) - c (v) ∣,$ and the right-hand side tends to $0$ because $c (v_{k})$ is bounded by step 1.1, $v_{k} \to v$ (step 2.1) and $c (v_{k}) \to c (v)$ (step 3.1) [F8]. As $\exp_{p}$ is smooth, hence continuous, on $E_{p}$ [F7], $\exp_{p} (w_{k}) \to \exp_{p} (w)$ ; but $\exp_{p} (w_{k}) = q_{k} \to q$ (steps 1.1 and 2.1), so $q = \exp_{p} (w) = \exp_{p} (c (v) v) = γ_{v} (c (v))$ by $γ_{v} (t) = \exp_{p} (t v)$ [F1].

5.1F1F9givenstep 4.1

Conclusion: the cut locus is closed. [F1, F9, given, step 4.1] Step 4.1 exhibits the limit $q$ as $\exp_{p} (c (v) v)$ with $v \in S_{p} M$ a unit vector and $c (v) = L < + \infty$ ; therefore $q \in Cut (p)$ by the definition of the cut locus [F1]. Since the convergent sequence $(q_{k})$ in $Cut (p)$ was arbitrary, $Cut (p)$ is sequentially closed, and by the metric-space equivalence between closedness and sequential closedness [F9] it is closed in $(M, d_{g})$ .

6.1A1F1F4F5F6F9givenstep 1.1step 3.1step 4.1step 5.1

Boundary and choice audit. [A1, F1, F4, F5, F6, F9, given, step 1.1, step 3.1, step 4.1, step 5.1] In dimension zero $T_{p} M = {0_{p}}$ , so $∣ 0_{p} ∣_{g} = 0 \neq 1$ and $S_{p} M = \emptyset$ ; hence $Cut (p) = \emptyset$ by [F1], and the empty set is closed in $(M, d_{g})$ [F9], both because its complement $M$ is open and because the sequential criterion holds vacuously. The empty manifold has no point $p$ . In dimension one the unit sphere has exactly two points and every step applies verbatim, the sequential compactness [F4] being immediate for a two-point set. Since $c$ takes values in $(0, + \infty]$ [F5], the limiting time $c (v) = L$ of step 3.1 is positive, so the degenerate value $L = 0$ (which would force the cut point to be the base point $p$ ) does not occur, and all directions occurring above are unit vectors, so no constant geodesic arises. Completeness enters only through the global exponential domain [F6] referenced in step 4.1 and through the cut-time continuity theorem [F5]; no compactness of $M$ is assumed, the compactness used being that of the finite-dimensional sphere $S_{p} M$ [F4]. The choice audit: ${A C}_{ω}$ of [A1] is spent exactly twice, once in step 1.1, where one direction $v_{k}$ is selected from each of the countably many nonempty fibres over the $q_{k}$ , and once in step 5.1 through the direction of [F9] that converts sequential closedness into closedness (contrapositively, one point is selected from each of the countably many nonempty sets $B (q, 1 / (n + 1)) \cap Cut (p)$ ).

$□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed pp.173-190, develops the cut locus of a point; Datar, Lectures on Riemannian Geometry, Lecture 23, sections 23.2-23.3, printed pp.163-172, presents the cut locus and the cut time. The closedness of $Cut (p)$ is proved above as sequential closedness, from the continuity of the cut time and the compactness of the finite-dimensional unit sphere; no source text is quoted.

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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 (2025) (standard reference, not scraped)