In this article, we study local holomorphic isometric embeddings from Bn into BN1 × · · · × BNm 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 and 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.