A Quantitative Closing Lemma and Partner Orbits on Riemannian Manifolds with Negative Curvature

The paper proves a quantitative closing lemma for geodesic flows on compact pinched negatively curved manifolds with explicit constants and shadowing, and the candidate solution proves the same result with the same constants but via a different (contact Anosov + Sasaki-metric) route. The paper’s Section 3 constructs axes of deck transformations and derives the bounds geometrically on the universal cover, yielding |T−T′| ≤ 2Cδ and d1(φs w, φs u) ≤ (5C+1)δ with C comparable to (4π/κ1)(2κ2/κ1+3) (Theorem 1 and Theorem 3.14), while the candidate solution obtains the same constants from a metric-comparison estimate and a Lipschitz bound on temporal distance for contact Anosov flows. A minor presentational mismatch is that the introduction states C = (4π/κ1)(2κ2/κ1+3) (see Theorem 1) whereas Theorem 3.14 records C = (2/κ1)(2κ2/κ1+3)·max{2π, κ1}; the candidate’s approach attains the sharper 4π/κ1 form uniformly. This is a difference in proof strategies and constants packaging rather than a correctness issue.