I need some help with this exercise.
Let $g:]-\pi/2,\pi/2[ \to R^2, \Theta \to \sin(\Theta)\cos(\Theta)(\cos(\Theta),\sin(\Theta))$.
Now let $M$ be the image of $g$.
I'm asked to decide, wether M is a submanifold or not.
There is a hint which told me to look how many components a small enough neighbourhood of $(0,0)$ in $M$ without $(0,0)$ has.
The answer should be two, as i can write the neigbourhood as a union of the $(x,y)$ which are right of $(0,0)$ and the values $(-x,y)$ which are at the opposite site.
I'm not sure how to procceed, has anyone (another) hint for me?
Thanks in advance
Umm,intervals are connected, continuous images of connected sets are connected so certainly this is the case for $\mathcal{C}^K$ maps. If you assume $M = g(X = (-\pi/2, \pi/2))$ is a sub-manifold then your map $g: X \to M \subset \mathbb{R}^2$ is a local diffeomorphism and so by invariance of domain,
$$\textbf{dim}(M) = \textbf{dim}(X) = 1$$
Hence, $g$ should be a local diffeomorphism onto $(0,0)$. You showed that $M$ has two components at $(0,0)$ i.e you can think of $M = (a,b) \cup (b,c)$. Thus, $g|_{X':=X \setminus g^{-1}(0,0)} : X' \to M\setminus \{(0,0)\} $ is not a local diffeomorphism, but it should be since it is just a restriction.