I know that isomorphism has a well known symbol, but what about diffeomorphism?
What's the symbol for diffeomorphism?
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$\begingroup$
functions
notation
smooth-manifolds
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0Well, since it is an isomorphism, why not use that symbol? – 2017-02-17
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0I prefer the notation $\cong$ that Zev mentions in his answer. Lee's *Introduction to Smooth Manifolds* introduces the notation $\approx$ but only seems to use it occasionally. – 2017-02-17
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0I'm not sure I would agree that "isomorphism has a well known symbol". Whatever symbol you decide to use in whatever you are writing, make sure that you say clearly say in words what your symbol means. Those, like me, who do not know what is "well known" will appreciate it. – 2017-02-19
1 Answers
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The usual symbol $\cong$ is fine, e.g. if $M$ and $N$ are smooth manifolds, then you can write $M\cong N$ for saying "the manifolds $M$ and $N$ are diffeomorphic" and $f:M\xrightarrow{\;\cong\;}N$ for saying "the map $f$ is a diffeomorphism from $M$ to $N$". After all, diffeomorphisms are just the isomorphisms in the category of smooth manifolds.