Geometric explanation of strong trajectory separation

Determine whether strong separation of neighboring fluid trajectories in the pressure-induced effective curved space arises from a local alignment of the velocity vector with an eigenvector of the Riemann curvature tensor, such that the particle closely shadows a particular geodesic while neighboring particles do not.

Background

The paper identifies attracting and repelling Lagrangian coherent structures with convergence and divergence of geodesics in the effective space generated by the pressure field. It relates finite-time Lyapunov exponent ridges to large eigenvalues of the Riemann curvature tensor projected along the geodesic flow.

The proposed explanation for enhanced relative separation is explicitly labeled an interpretive conjecture. The paper presents it as consistent with earlier observations about least-stretching geodesics, but does not prove the claimed alignment mechanism.

References

Building upon this picture, strong separation may arise from a parallelism effect. When the local velocity vector aligns with an eigenvector of the Riemann tensor, the particle closely shadows a particular geodesic, so that its deviation from that geodesic is reduced, whereas neighboring particles do not. As a result, the relative separation between them is enhanced. Such an interpretive conjecture is consistent with the observation of \citet{haller2012geodesic} that the locally most compressing or most shearing material lines are closely shadowed by least-stretching geodesics of the metric induced by the Cauchy-Green strain tensor.

— On the relationships between pressure geometrization and vortex identification  (2610.11980 - Tu et al., 8 Oct 2026) in Section 4.4, Finite-time Lyapunov exponent