The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a Calabi-Eckmann manifold. If $f,g \colon V_n \longrightarrow \mathbb{P}^n$ are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: $f(x)=g(x)$ for some $x\in V_n$. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form $G/P$ where $G$ is complex simple algebraic group and $P\subset G$ is a maximal parabolic subgroup, where one of the maps is dominant.

Let $B_1$ and $B_2$ be the open unit balls of ${\bbd C}^{n_1}$ and ${\bbd C}^{n_2}$ for the norms $\Vert\,{.}\,\Vert_1$ and $\Vert\,{.}\, \Vert_2$. Let $f \colon B_1 \rightarrow B_2$ be a holomorphic mapping such that $f(0)=0$. It is well known that, for every $z \in B_1$, $\Vert f(z)\Vert_2 \leq \Vert z \Vert_1$, and $\Vert f'(0)\Vert \leq 1$. In this paper, I prove the converse of this result. Let $f \colon B_1 \rightarrow B_2$ be a holomorphic mapping such that $f'(0)$ is an isometry. If $B_2$ is strictly convex, I prove that $f(0) =0$ and that $f$ is linear. I also define the rank of a point $x$ belonging to the boundary of $B_1$ or $B_2$. Under some hypotheses on the ranks, I prove that a holomorphic mapping such that $f(0) = 0$ and that $f'(0)$ is an isometry is linear.