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 holomorphic continuation and analyzing the real analytic subvariety carefully, we show that a component is either constant or proper holomorphic map between balls. Hence the total geodesy of non-constant components follows from a linearity criterion of Huang. This is a joint work with Y. Zhang.