Math 6510 • Differentiable Manifolds

True or false (provide justification): Let \(M\subset\mathbb{R}^n\) be a smooth
submanifold of dimension \(m\). Then there are \(n-m\) smooth functions
\(f_1,\ldots,f_m:\mathbb{R}^n\to\mathbb{R}\) such that:
\begin{enumerate}
\item \(M=\{x\in\mathbb{R}^n:f_1(x)=\cdots=f_{n-m}(x)=0\}\).
\item For each \(x\in M\) the gradients
\(\nabla(f_1)(x),\ldots,\nabla(f_{n-m})(x)\) are linearly independent.
\end{enumerate}