  This work examines the magnetic order and spin dynamics of a double-exchange
model with competing ferromagnetic and antiferromagnetic Heisenberg
interactions between the local moments. The Heisenberg interactions are
periodically arranged in a Villain configuration in two dimensions with
nearest-neighbor, ferromagnetic coupling $J$ and antiferromagnetic coupling
$-\eta J$. This model is solved at zero temperature by performing a
$1/\sqrt{S}$ expansion in the rotated reference frame of each local moment.
When $\eta $ exceeds a critical value, the ground state is a magnetically
frustrated, canted antiferromagnet. With increasing hopping energy $t$ or
magnetic field $B$, the local moments become aligned and the ferromagnetic
phase is stabilized above critical values of $t$ or $B$. In the canted phase, a
charge-density wave forms because the electrons prefer to sit on lines of sites
that are coupled ferromagnetically. Due to a change in the topology of the
Fermi surface from closed to open, phase separation occurs in a narrow range of
parameters in the canted phase. In zero field, the long-wavelength spin waves
are isotropic in the region of phase separation. Whereas the average spin-wave
stiffness in the canted phase increases with $t$ or $\eta $, it exhibits a more
complicated dependence on field. This work strongly suggests that the jump in
the spin-wave stiffness observed in Pr$_{1-x}$Ca$_x$MnO$_3$ with $0.3 \le x \le
0.4$ at a field of 3 T is caused by the delocalization of the electrons rather
than by the alignment of the antiferromagnetic regions.
