The problem : it is a submanifold ?
My works : I tried to write this as the following set :
$$ X = \left\{ (\theta,r) \mbox{ : } r - ab^{\theta} = 0 \mbox{, } (a,b) \in \left( \mathbb{R}^{+*} \right)^2 \right\} $$
but X cannot be the curve on the picture because there are "dots"... and I don't know how to write it.
Intuitively, it is not a submanifold but can you give me some indications to show it ?
Thanks you.