  It's well-known in \kahler geometry that the infinite dimensional symmetric
space $\hcal$ of smooth \kahler metrics in a fixed \kahler class on a polarized
\kahler manifold is well approximated by finite dimensional submanifolds
$\bcal_k \subset \hcal$ of Bergman metrics of height $k$. Then it's natural to
ask whether geodesics in $\hcal$ can be approximated by Bergman geodesics in
$\bcal_k$. For any polarized \kahler manifold, the approximation is in the
$C^0$ topology. While Song-Zelditch proved the $C^2$ convergence for the
torus-invariant metrics over toric varieties. In this article, we show that
some $C^{\infty}$ approximation exists as well as a complete asymptotic
expansion for principally polarized Abelian varieties. We also get a $C^\infty$
complete asymptotic expansion for harmonic maps into $\bcal_k$ which
generalizes the work of Rubinstein-Zelditch on toric varieties.
