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.
EDIT : I have to give a geometric proof.