Let $f:{\mathtt R^n} \to {\mathtt R^n}$ be a smooth map and $df(x)$ is nonsingular for all $x$, in addition, $\mathop {\lim }\limits_{\left| x \right| \to + \infty } \left| {f(x)} \right| = + \infty $, then is it necessarily that $f$ is injective? |
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Because $f$ is proper and locally diffeomorphic, $f: \mathtt {R^n} →\mathtt {R^n}$ is an universal covering map. Since $\mathtt {R^n}$ is simply-connected, then the deck transformation group is trivial and therefore $f$ is injective. The same method can be applied to the general theorem of Hadamard |
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