It is shown that the Eshelby inclusions, that is inclusions of uniform eige
nstrains and constant eigenstress, form in three dimensions a connected nin
e-dimensional manifold, and, as a consequence, the only perturbations of an
ellipsoid that preserve the Eshelby property are into another ellipsoid. T
he method is illustrated in both two and three dimensions, and can be also
used to treat similar questions of other than constant dependence of the st
ress.