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


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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.