In a linear space, say $\mathbb{R}^{d}$, I know we can define embedded submanifolds with a locally defining function, which is equivalent to the canonical definition via diffeomorphisms. Following is the definition found in An introduction to optimization on manifolds
Definition 3.10. Let $\mathcal{E}$ be a linear space of dimension d. A non-empty subset $\mathcal{M}$ of $\mathcal{E}$ is a (smooth) embedded submanifold of $\mathcal{E}$ of dimension $n$ if either
- $n=d$ and $\mathcal{M}$ is open in $\mathcal{E}$-we also call this an open submanifold; or
- $n=d-k$ for some $k \geq 1$ and, for each $x \in \mathcal{M}$, there exists a neighborhood $U$ of $x$ in $\mathcal{E}$ and a smooth function $h: U \rightarrow \mathbb{R}^k$ such that (a) If $y$ is in $U$, then $h(y)=0$ if and only if $y \in \mathcal{M}$; and (b) $\operatorname{rank} \mathrm{D} h(x)=k$. Such a function $h$ is called a local defining function for $\mathcal{M}$ at $x$.
I wonder if we already have a locally defining function, in fact in my case, a globally defining function, but the rank of $Dh(x)$ is not the same in all $x$, does that basically rules out the possibility that the set I'm referring to is an embedded submanifolds?
I know some authors accept something like the disjoint union of two manifolds with different dimensions as a manifold, but I'm not sure that's relevant or useful in my case. For me, I need to consider a space in $\mathbb{R}^{n}$, and all points such that $\sum_{i}(p_{i}(x))^{2} = 0$, where $p_{i}(x):\mathbb{R}^{n} \to \mathbb{R}$ is a quadratic function that does not use all entries in $\mathbb{R}^{n}$. I can technically break this globally defining function into multiple parts (since it is a sum of squares) such that it maps to $\mathbb{R}^{k}$ for some $k$.
Sorry for not providing the exact details as this is related to my research, and I'm not working alone.