  MHz conductivity, torque magnetometer and magnetization measurements are
reported on single crystals of CeOs$_4$Sb$_{12}$ and NdOs$_4$Sb$_{12}$ using
temperatures down to 0.5~K and magnetic fields of up to 60~tesla. The
field-orientation dependence of the de Haas-van Alphen and Shubnikov-de Haas
oscillations is deduced by rotating the samples about the $[010]$ and
$[0\bar{1}1]$ directions. The results indicate that NdOs$_4$Sb$_{12}$ has a
similar Fermi surface topology to that of the unusual superconductor
PrOs$_4$Sb$_{12}$, but with significantly smaller effective masses, supporting
the importance of local phonon modes in contributing to the low-temperature
heat capacity of NdOs$_4$Sb$_{12}$. By contrast, CeOs$_4$Sb$_{12}$ undergoes a
field-induced transition from an unusual semimetal into a high-field,
high-temperature state characterized by a single, almost spherical
Fermi-surface section. The behavior of the phase boundary and comparisons with
models of the bandstructure lead us to propose that the field-induced phase
transition in CeOs$_4$Sb$_{12}$ is similar in origin to the well-known
$\alpha-\gamma$ transition in Ce and its alloys.
