  In this article, we study local holomorphic isometric embeddings from
${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the
normalized Bergman metrics up to conformal factors. Assume that each conformal
factor is smooth Nash algebraic. Then each component of the map is a
multi-valued holomorphic map between complex Euclidean spaces by the algebraic
extension theorem derived along the lines of Mok and Mok-Ng. Applying
holomorphic continuation and analyzing real analytic subvarieties carefully, we
show that each component is either a constant map or a proper holomorphic map
between balls. Applying a linearity criterion of Huang, we conclude the total
geodesy of non-constant components.
