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arxiv: 2605.26499 · v1 · pith:WNMYEQKYnew · submitted 2026-05-26 · 🧮 math.DG

Stability of the Injectivity Radius and the Cut Locus of Submanifolds under Perturbations

Aritra Bhowmick This is my paper

Pith reviewed 2026-06-29 16:16 UTC · model grok-4.3

classification 🧮 math.DG
keywords injectivity radiuscut locussubmanifoldsRiemannian metricHausdorff stabilityWhitney topologyC2 perturbationsnormal geodesics
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The pith

The injectivity radius of a compact submanifold varies continuously under C² perturbations of the Riemannian metric, with its cut locus remaining Hausdorff stable even under Whitney C² changes to the submanifold itself.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends earlier continuity results for the injectivity radius, first proved for compact manifolds and then for points, to the setting of compact submanifolds. It establishes that the injectivity radius function remains continuous when the metric is varied in the C² topology. The same continuity yields Hausdorff stability of the cut locus under simultaneous C² perturbations of the metric and, optionally, of the submanifold in the Whitney sense. A reader cares because these properties let one deform submanifolds or metrics while keeping control over the first point where normal geodesics stop being minimizing.

Core claim

We first show that the injectivity radius of a submanifold depends continuously on the metric. Then, we obtain the Hausdorff stability of the cut locus of the submanifold, under C² perturbation of the metric. In fact, we allow the submanifold to be perturbed in the Whitney C² sense as well.

What carries the argument

The injectivity radius of the submanifold, which is the infimum along normal geodesics of the distance to the first cut point, and the cut locus itself viewed as a set in the ambient manifold.

If this is right

  • The cut locus of the submanifold varies continuously in the Hausdorff metric whenever the ambient metric is perturbed in the C² topology.
  • Hausdorff stability of the cut locus continues to hold when the submanifold itself is varied simultaneously in the Whitney C² topology.
  • Both the continuity of the injectivity radius and the Hausdorff stability of the cut locus generalize the statements previously known only for points.
  • The proofs adapt the original arguments of Ehrlich and Sakai by replacing pointwise geodesic analysis with normal exponential maps from the submanifold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same continuity might be used to track how the medial axis or focal set of a submanifold moves under small deformations of the metric.
  • Numerical schemes that approximate geodesics from a submanifold could exploit the stability to justify replacing a given metric by a nearby smoother one without large changes in the cut locus.
  • The result suggests that analogous stability statements may hold for other first-order geometric objects defined via the normal exponential map, such as the distance function to the submanifold.

Load-bearing premise

The submanifold is required to be compact and all perturbations are restricted to the C² category.

What would settle it

A sequence of C²-close Riemannian metrics on a fixed compact submanifold for which the injectivity radius fails to converge to its original value.

read the original abstract

The continuity of the injectivity radius of a compact manifold under $C^2$ perturbation of the Riemannian metric was originally proved by P. Ehrlich (Composito Math., 1974), and later the proof was simplified by T. Sakai (Math. J. Okayama Univ., 1983). Using this continuity, jointly with J. Itoh and S. Prasad (J. Math. Anal. Appl., 2025), we proved the Hausdorff stability of the cut locus of a point, when both the point and the metric are perturbed. In the present article, we extend both these results to submanifolds. We first show that the injectivity radius of a submanifold depends continuously on the metric. Then, we obtain the Hausdorff stability of the cut locus of the submanifold, under $C^2$ perturbation of the metric. In fact, we allow the submanifold to be perturbed in the Whitney $C^2$ sense as well.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper extends the C^2 continuity of the injectivity radius (originally due to Ehrlich and Sakai for points on compact manifolds) to compact submanifolds, and proves Hausdorff stability of the cut locus of a submanifold under C^2 metric perturbations, allowing also Whitney C^2 perturbations of the submanifold itself.

Significance. If the claims hold, the results generalize classical stability theorems to the submanifold setting with a natural strengthening via Whitney perturbations; this could support further work on geometric invariants and variational problems involving submanifolds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for acknowledging the potential significance of extending the classical C^2 continuity results of Ehrlich and Sakai to the setting of compact submanifolds, including the allowance for Whitney C^2 perturbations. No specific major comments were provided in the report, so we have no individual points to address at this time. We are happy to provide further clarifications or revisions if the referee has additional questions.

Circularity Check

0 steps flagged

Minor self-citation to prior joint work; extension to submanifolds remains independent

full rationale

The derivation extends the Ehrlich/Sakai continuity result and the authors' own prior joint result on points (cited as Bhowmick-Itoh-Prasad 2025) to the submanifold setting via direct arguments under the same C^2 and compactness hypotheses. No equation or claim reduces by definition or fitting to its own inputs, and the self-citation supports only the base point case rather than carrying the new submanifold claims. This matches the expected minor non-load-bearing self-reference pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The work inherits compactness and C^2 regularity assumptions from the referenced point-case theorems.

axioms (2)
  • domain assumption The ambient manifold and submanifold are compact.
    Required for the original continuity result on points and carried over to the submanifold setting.
  • domain assumption Perturbations are of class C^2 (metric) or Whitney C^2 (submanifold).
    Stated as the regularity class under which the stability holds.

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discussion (0)

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Reference graph

Works this paper leans on

19 extracted references · 14 canonical work pages

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