  X-ray observations of galaxy clusters have detected numerous X-ray cavities,
evolved from the interaction of AGN jets with the intracluster medium (ICM) and
providing compelling evidence for the importance of jet-mode AGN feedback. Here
we argue for the physical importance of the cavity shape, which we characterize
with two geometric parameters: radial elongation $\tau$ and top wideness $b$.
We study the cavity shape with 16 hydrodynamic jet simulations in two
representative clusters, and find that the shapes of young cavities are mainly
determined by various jet properties. Our simulations successfully reproduce
two observed types of young cavities elongated along either the jet ($\tau>1$;
type-II) or perpendicular ($\tau\leq1$; type-I) direction. Bottom-wide type-I
cavities are produced by very light internally-subsonic jets, while top-wide
type-II cavities are produced by heavier, internally-supersonic jets, which may
also produce center-wide cavities with $\tau\sim 1$ if the jets are only
slightly supersonic. Bottom-wide type-II cavities can be produced by very light
jets with very long durations and cylindrical cavities are produced by very
light internally-supersonic jets. While not appreciably affecting the shapes of
young cavities, viscosity significantly affects the long-term cavity evolution,
suppressing both interface instabilities and the formation of torus-like
morphology. We encourage observers to study the shapes of young and old X-ray
cavities separately, the former probing the properties of AGN jets and the
latter potentially probing the ICM viscosity level.
