  We study the possibility to explain the mystery of the dark matter through
the transition from General Relativity to embedding gravity. This modification
of gravity, which was proposed by Regge and Teitelboim, is based on a simple
string-inspired geometrical principle: our spacetime is considered here as a
4-dimensional surface in a flat bulk. We show that among the solutions of
embedding gravity, there is a class of solutions equivalent to solutions of GR
with an additional contribution of non-relativistic embedding matter, which can
serve as cold dark matter. We prove the stability of such type of solutions and
obtain an explicit form of the equations of motion of embedding matter in the
non-relativistic limit. According to them, embedding matter turns out to have a
certain self-interaction, which could be useful in the context of solving the
core-cusp problem that appears in the LambdaCDM model.
