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Let $\Omega \subset \mathbb R^3$ be a bounded regular simply connected domain contained in a ball $S$. Assume also that $\Omega$ is simply connected by surfaces (i.e. every regular closed surface contained in $\Omega$ can be continuously deformed shrinking it to a point).

Is it possible to define a $C^\infty(\overline{S \setminus \Omega})$ field of unit vectors that extends the external normal vectors to $\partial \Omega$ to the entire domain $\overline{S\setminus \Omega}$ ending, on the boundary $\partial S$ in the external normal vectors to $\partial S$?

Also, can we prove this under simpler/weaker assumptions too?

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  • $\begingroup$ Do you mean a ball $S$? $\endgroup$
    – Jochen Wengenroth
    Commented Jul 20, 2021 at 13:25
  • $\begingroup$ @JochenWengenroth Yes $\endgroup$
    – user934318
    Commented Jul 23, 2021 at 15:53

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