# differential geometry – Show that a dashed curve is not a submanifold

I have to show that the following curve is not a submanifold. I tried to define this as a set, as the spiral logarithm, but the “dots” block me..

My definition of a submanifold is : $$M$$ is a submanifold iff

$$forall x in M mbox{, } exists U subset mathbb{R}^2 mbox{, } exists f : U rightarrow mathbb{R} mbox{, } M cap U = f^{-1} left( left{ 0 right} right)$$

where $$U$$ is a neighborhood of $$x$$ and $$f$$ a submersion in $$x$$.

Thanks you.