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Otis Chodosh
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Is there a $C^\infty$-smooth mapembedding $\gamma : I \to \mathbb{R}^3$ with $\gamma'(t) \neq 0$ for all $t$ so so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?

Is there a $C^\infty$-smooth map $\gamma : I \to \mathbb{R}^3$ with $\gamma'(t) \neq 0$ for all $t$ so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?

Is there a $C^\infty$-smooth embedding $\gamma : I \to \mathbb{R}^3$ so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?

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YCor
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Otis Chodosh
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Smooth curve in $\mathbb{R}^3$ not contained in real analytic surface?

Is there a $C^\infty$-smooth map $\gamma : I \to \mathbb{R}^3$ with $\gamma'(t) \neq 0$ for all $t$ so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?