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 $\gamma(I)\subset M$?
Smooth curve in $\mathbb{R}^3$ not contained in real analytic surface?
Otis Chodosh
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