  In this study, the dynamics of a dissipationless incompressible Hall
magnetohydrodynamic (HMHD) medium are formulated as geodesics on a direct
product of two volume-preserving diffeomorphism groups. Formulations are given
for the geodesic and Jacobi equations based on a linear connection with
physically desirable properties, which agrees with the Levi-Civita connection.
Derivations of the explicit normal-mode expressions for the Riemannian metric,
Levi-Civita connection, and related formulae and equations are also provided
using the generalized Els\"asser variables (GEVs). Examinations of the
stabilities of the hydrodynamic (HD, $\alpha=0$) and magnetohydrodynamic (MHD,
$\alpha\to0$) motions and the $O(\alpha)$ Hall-term effect in terms of the
Jacobi equation and the Riemannian sectional curvature tensor are presented,
where $\alpha$ represents the Hall-term strength parameter. It is very
interesting that the sectional curvatures of the MHD and HMHD systems between
two GEV modes were found to take both the positive (stable) and negative
(unstable) values, while that of the HD system between two complex helical
waves was observed to be negative definite. Moreover, for the MHD case,
negative sectional curvatures were found to occur only when mode interaction
was "local," i.e., the wavenumber moduli of the main flow (say $p$) and
perturbation (say $k$) were close to each other ($k \approx p$). However, in
the nonlocal limit ($k \gg p$ or $k \ll p$), the sectional curvatures were
always positive. This result leads to the conjecture that the MHD interactions
mainly excite wavy or non-growing motions; however, some local interactions
cause dynamical instability that leads to chaotic or turbulent plasma motions.
Additionally, it was found that the tendencies of the $O(\alpha)$ effects are
opposite between the ion cyclotron and whistler modes. Comparison with
energy-Casimir method is discussed.
