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Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique? It is clear that the set of all such embeddings con …

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 $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ …

Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly? What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such th …

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 …