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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?

J. W. Tanner
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malmath
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  • You should correctly say that for each in $M$ there exists an open neighborhood $U$ and a diffeomorphism $f : U \to V \subset \mathbb R^n$ onto an open $V \subset \mathbb R^n$ such that ... – Paul Frost Mar 07 '23 at 09:16

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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 $$ \{(x_1,...,x_n)\in \mathbb R^n:x_{d+1}=...=x_n=0\} $$

Boxonix
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