I will talk about the rigidity for a local holomorphic isometric embedding
from ${\BB}^n$ into ${\BB}^{N_1} \times\cdots \times{\BB}^{N_m}$ with
respect to the normalized Bergman metrics. Each component of the map is a
multi-valued holomorphic map between complex Euclidean spaces by Mok's
algebraic extension theorem. By using the method of the holomorphic
continuation and analyzing real analytic subvarieties carefully, we show
that a component is either a constant map or a proper holomorphic map
between balls. Hence the total geodesy of non-constant components follows
from a linearity criterion of Huang. In fact, the rigidity is derived in a
more general setting for a local holomorphic conformal embedding. This is
a joint work with Y. Zhang.