Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item Show that the Klein bottle cannot be embedded (as a smooth submanifold) into
\(\mathbb{R}^3\).
\item It is a fact that the Klein bottle can be embedded into \(\mathbb{R}^4\). Show
that no such embedding \(K\subset\mathbb{R}^4\) can be given by a system of equations
\[
  f_1=0,\qquad f_2=0
\]
with independent smooth functions \(f_1,f_2\) (i.e. where \(df_1,df_2\) are linearly
independent at each point \(x\in K\)).
\end{enumerate}