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Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations: \begin{equation} P_1(\vec x)=0, \\ \vdots \\ P_m(\vec x)=0, \end{equation} (where $\ …

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} (where $\vec x$ are …

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} where $\vec x$ are c …

Thanks Willie but I find your answer slightly hard to follow. The following is what I uncovered. Given a basis $\{e_a\}$ (with $a=1,...,n-m$) for the tangent space $TM$ the second fundamental form, $\ …

Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$. Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ …

Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$? $X$ may have points at which the Jacob …