GEOMETRIC CALCULATIONS ON DENSITY MANIFOLDS FROM RECIPROCAL RELATIONS IN HYDRODYNAMICS

The paper explicitly proves the curvature identity for hydrodynamical density manifolds via the Eulerian Koszul route, giving the commutator formula, the Levi–Civita connection, and the three-block curvature expression (Γχ″-block, nested Γ-block, and the Δχ†-commutator block) in Theorem 2; the candidate solution follows the same strategy and arrives at the identical three-line formula. Minor differences are notational (e.g., an intermediate connection pairing the paper simplifies away) and a small sign/convention slip in the model’s metric-pairing remark that does not affect the final result. See the commutator (eq. (10)), the connection (eqs. (12) and (19)), and the curvature identity (eq. (24)) in the paper for direct confirmation .