Suppose we have the mapping $G:\mathbb{R}^2\rightarrow\mathbb{R}^4$ defined by $$ G(x,y)=\left((r \cos y + a)\cos x,(r\cos y + a)\sin x, r\sin y \cos \frac x2,r\sin y\sin \frac x2\right)$$ where $(x,y)\in \mathbb{R}^2$. Show that $G$ induces an embedding of Klein bottle into $\mathbb{R}^4$.
What I have tried:
I try to specify the kernel of $G$, so that the quotient space $\mathbb{R}^2/\ker(G)$ is a Klein bottle by observing the equivalence relationship. And we can show that $G$ is an embedding simply by proving $G$ is injective and $\mathbb{R}^2/\ker(G)$ is compact. But I stuck on figuring out the Kernel of $G$, where I've certainly messed things up. Could someone please show me how to do it? Thanks very much.
