Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] Define \(F:\R^4\to\R^2\) by
\[
  F(x_1,x_2,x_3,x_4)
  =(x_1^2+x_2^2+x_3^2+x_4^2,\ x_1^2+x_2^2-x_3^2-x_4^2).
\]
Show that \(M=F^{-1}(1,0)\) is a smooth manifold.
\item[(b)] For each \(x=(x_1,x_2,x_3,x_4)\in M\) show that the tangent space
\(T_xM\) is spanned by \((x_2,-x_1,0,0)\) and \((0,0,x_4,-x_3)\).
\item[(c)] Let \(G:\R^4\to\R\) be a smooth map and \(g=G|_M\) the restriction of
\(G\) to \(M\). Show that \(x\) is a critical point of \(g\) if and only if
\(\ker F\subset\ker G\).
\item[(d)] If \(G(x_1,x_2,x_3,x_4)=x_1+x_3\) find the critical points of \(g\).
\end{enumerate}