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

Rigidity for local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ up to conformal factors