  In this paper, we use the canonical connection instead of Levi-Civita
connection to study the smooth maps between almost Hermitian manifolds,
especially, the pseudoholomorphic ones. By using the Bochner formulas, we
obtian the $C^2$-estimate of canonical second fundamental form, Liouville type
theorems of pseudoholomorphic maps, pseudoholomorphicity of pluriharmonic maps,
and Simons integral inequality of pseudoholomorphic isometric immersion.
