In some books, in the definition of submanifolds, they write \ "Let $M$ be a submanifold of $\mathbb{R^{n}}$ of dimension d if for every $x\in M$ there exists an open neighborhood such that. $$f(U\cap M)=f(U)\cap \mathbb{R^{d}}$$ and in other book \ Let $M$ be a submanifold of $\mathbb{R^{n}}$ of dimension d if for every $x\in M$ there exists an open neighborhood such that. $$f(U\cap M)=f(U)\cap (\mathbb{R^{d}}\times\{0\})$$ My question is, what is the difference between these two definitions?
More precisely, why in the second definition do we add the zero and is there a difference?
The first book simply omits the zero, assuming that it's the "logical" way of thinking of $\mathbb R^d$ as a subset of $\mathbb R^n$. The second definition could be thought of as more precise, where you explicitly state $0\in\mathbb R^{n-d}$ to make sure you're talking about the set $$ \{(x_1,...,x_n)\in \mathbb R^n:x_{d+1}=...=x_n=0\} $$