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A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$

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  • Using the half-Reeb component technique as introduced in [10], we try to clarify the intrinsic relation between the injectivity of differentiable local homeomorphisms $X$ of $R^2$ and the asymptotic behavior of real eigen-values of derivations $DX(x)$. The main result shows that a differentiable local homeomorphism $X$ of $R^2$ is injective and that its image $X(R^2)$ is a convex set if $X$ satisfies the following condition: (*) There does not exist a sequence $R^2$ ∋ $x_i\rightarrow \infty$ such that $X(x_i)\rightarrow a\in \R^2$ and $DX(x_i)$ has a real eigenvalue $\lambda _i\rightarrow 0$. When the graph of $X$ is an algebraic set, this condition becomes a necessary and sufficient condition for $X$ to be a global diffeomorphism.
    Mathematics Subject Classification: Primary: 14R15; Secondary: 14E07, 14E09, 14E40.


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