  Using the symmetry theory we analyze of the flexomagnetic effect in all 90
magnetic classes and showed that 69 of them are flexomagnetic. Then we explore
how the symmetry breaking, inevitably present in the vicinity of the surface,
changes the local symmetry and thus the form of the flexomagnetic tensors. All
possible surface magnetic classes (in the number of 19) were obtained from the
90 bulk magnetic classes for the surface cuts 001, 010 and 100 types. It
appeared that all 90 bulk magnetic classes become flexomagnetic, piezomagnetic
and piezoelectric in the vicinity of surface. Using the free energy approach,
we show that the flexomagnetic effect leads to a new type of
flexo-magnetoelectric (FME) coupling in nanosized and bulk materials, in all
spatial regions, where the polarization and (anti)magnetization vectors are
spatially inhomogeneous due to external or internal forces. The linear FME
coupling, proportional to the product of the gradients of (anti)magnetization
and polarization, flexoelectric and flexomagnetic tensors, is significant in
nanosized ferroelectrics-(anti)ferromagnetics, where gradients of the
polarization and magnetization obligatory exist. The spontaneous FME coupling
induced by the spatial confinement give rise to the size-dependent linear
magnetoelectric coupling in nanosized ferroelectrics-(anti)ferromagnetics.
  We show that the flexomagnetic effect may lead to improper ferroelectricity
in bulk (anti)ferromagnetics via the linear and nonlinear FME coupling.
Inhomogeneous spontaneous polarization is induced by the (anti)magnetization
gradient, which exists in all spatial regions, where polarization varies and
(anti)magnetization vector changes its direction. The gradient can be induced
by the surface influence as well as by external strain via e.g. the sample
bending.
