A Characteristic Property of Injective Holomorphic Germs

Abstract.

Let X ₀ be the germ at 0 of a complex variety and let $f:\ X_0\rightarrow {\Bbb C}^n_0$ be a holomorphic germ. We say that f is pseudoimmersive if for any $g:\ {\Bbb R}_0\rightarrow X_0$ such that $f \circ g \in C^{\infty}$, we have $g\in C^{\infty}$. We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered.