Extension of Fujimoto's uniqueness theorems

Hirotaka Fujimoto considered two meromorphic maps $f$ and $g$ of ${\mathbb{C}}^m$ into ${\mathbb{P}}^n$ such that $f^{\ast }(H_j)=g^{\ast }(H_j)$ ($1\le j\le q$) for $q$ hyperplanes $H_j$ in ${\mathbb{P}}^n$ in general position and proved $f=g$ under suitable conditions. This paper considers the case where $f$ is into ${\mathbb{P}}^n$ and $g$ is into ${\mathbb{P}}^N$ and gives extensions of some of Fujimoto’s uniqueness theorems. The dimensions $N$ and $n$ are proved to be equal under suitable conditions. New and interesting phenomena also occur.