Curvature Rigidity Through Level Sets of Lyapunov Exponents in Geodesic Flows

The paper’s Theorem 1.3 asserts that if M has no conjugate points and curvature bounded below, and if an invariant measure μ satisfies μ(Λα)=1 with α ≥ √(−∫Ric dμ), then K is constant on π(Supp(μ)) and equals ∫Ric dμ; its proof uses the Riccati equation, a trace Cauchy–Schwarz bound, time-averaging, and equality forcing Uu to be a scalar multiple of the identity, which via Riccati gives constant curvature on π(Supp(μ)) . The candidate solution follows the same core route (Riccati + matrix Cauchy–Schwarz + invariance + equality), phrased through the global differential inequality d/dt f ≤ −f^2/m − m Ric, producing α^2 + ∫Ric ≤ 0 and then equality rigidity. The main gap in the candidate is not stating the paper’s needed hypotheses (no conjugate points and curvature bounded below) that ensure integrability/regularity for the manipulations with Uu and Fubini’s theorem. With these standard hypotheses added, the candidate’s proof matches the paper’s logic and conclusion.