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 $\gamma(I)\subset M$?



Smooth curve in $\mathbb{R}^3$ not contained in real analytic surface?
Here is an example: Let $\gamma:\mathbb{R}\to\mathbb{R}^3$ be defined by $\gamma(t) = \bigl(t,\exp(-1/t^2),0\bigr)$ for $t<0$, $\gamma(0) = (0,0,0)$ and $\gamma(t) = \bigl(t,0,\exp(-1/t^2)\bigr)$ for $t>0$. Then I claim that there is no nonsingular real-analytic surface $M\subset\mathbb{R}^3$ that contains the image of $\gamma$.


To see this, suppose that such an $M$ does exist and let $u\in S^2$ be a unit normal to $M$ at the origin. If $u\not=(0,0,\pm1)$, then $M$ and the plane $z=0$ intersect transversely at $(0,0,0)$ along a real-analytic curve that contains $\gamma\bigl((-\epsilon,0]\bigr)$ for some $\epsilon>0$, which is clearly impossible. Similarly, we reach a contradiction if $u\not=(0,\pm1,0)$. Thus, $M$ does not exist.


