Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] State the definition of a regular value and state the pre-image theorem.
\item[(b)] Let \(F:\R^n\to\R^m\) be a smooth function with \(y\in\R^m\) a regular value
and let \(M=F^{-1}(y)\). If \(f:\R^n\to\R^k\) is a smooth function and \(x\in M\) is
critical point for \(f|_M\) what are the possible values for the dimension of
\(\ker F_*(x)\cap\ker f_*(x)\)? (Here \(F_*(x)\) is the tangent map from
\(T_x\R^n\to T_y\R^m\) and \(f_*(x)\) is also a tangent map.)
\item[(c)] Define \(G:\R^n\to\R^{m+k}\) by \(G(x)=(F(x),f(x))\).
\begin{enumerate}
\item[(i)] If \(z\in\R^k\) is regular value of \(f|_M\) is
\((y,z)\in\R^{m+k}\) always a regular value of \(G\)? Give a proof or find a
counterexample.
\item[(ii)] If \(z\in\R^k\) is a regular of \(f\) (as a function on \(\R^n\)) is
\((y,z)\in\R^{m+k}\) always a regular value of \(G\). Give a proof or find a
counterexample.
\end{enumerate}
\end{enumerate}