In Topological methods in hydrodynamics 1 mentioned that "The Riemannian curvature of a manifold has a profound impact on the behavior of geodesics on it. If the Riemannian curvature of a manifold is positive (as for a sphere or for an ellipsoid), then nearby geodesics oscillate about one another in most cases, and if the curvature is negative (as on the surface of a one-sheet hyperboloid), geodesics rapidly diverge from one another."
I am curious, if the curvature is positive, but non-constant, does this also imply some kind of stability(eg: Jacobi stability, Lagrange stability or Lyapunov stability) of the geodesics?
I did some research and found that even positive curvature may lead to chaotic phenomena [2]. But I did not find, if it is positive curvature and non-constant, what is the stability of the geodesic? By the Riemann geometric comparison theorem, We can know that if the positive curvature is bounded, the Jacobi field equation is also bounded, and then there should be no chaos which is conflict with the result of [2](positive curvature may cause chaos). Where did I get it wrong? Am I misunderstanding the comparison theorem?
Example in [2]:
In reference[2] give the 2-dimensional Riemannian manifold(Hénon-Heiles (HH) model) as an example. The curvature tensor has a unique component $R_{1212}$ that is not zero.
The HH Hamiltonian is written in the form \begin{equation} H=\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}\right)+\frac{1}{2}\left(q_{1}^{2}+q_{2}^{2}\right)+q_{1}^{2} q_{2}-\frac{1}{3} q_{2}^{3} . \end{equation}
It is shown here that chaos in the Hénon-Heiles model stems from parametric instability due to positive curvature fluctuations along the geodesics (dynamical motions) of configuration space manifold.
The link between stability and curvature is mathematically expressed by the Jacobi field equation for geodesic spread \begin{align} & \frac{d^{2} \xi_{1}}{d s^{2}}+\frac{1}{2} \mathscr{R} \xi_{1}=0, \\ & \frac{d^{2} \xi_{2}}{d s^{2}}=0, \end{align}
If the solution $\xi_{1}(t)$ computed along a given trajectory remains bounded or grows at most linearly with time, then this trajectory is stable. If $\xi_{1}(t)$ grows exponentially with time, then the trajectory is chaotic, i.e., unstable with respect to any variation-even arbitrarily small-of coordinates and momenta operated at any given point of it.
So my question is, why does the positive curvature of an interval, which is non-constant, lead to chaos? This seems to violate the comparison theorem in Riemannian geometry. According to the theorem, the positive curvature is bounded, so $\xi_{1}(t)$ should also be bounded, so why would it grow exponentially? In other words, does the positive curvature of non-constant numbers also imply some kind of stability?
I also asked this question in the physics forum(https://physics.stackexchange.com/questions/764902/curvature-and-stability), but after thinking about it, it seems more relevant to math, so I'm asking it here as well.
1: Arnold, Vladimir I., and Boris A. Khesin. 2021. Topological Methods in Hydrodynamics. Springer Nature. IV. Differential Geometry of Diffeomorphism Groups in page 214.
[2]: Casetti, Lapo, Marco Pettini, and E. G. D. Cohen. 2000. “Geometric Approach to Hamiltonian Dynamics and Statistical Mechanics.” Physics Reports 337 (3): 237–341. https://doi.org/10/cmtfhz.