I am looking at the following set $$M:=\{(x,y) \in \Bbb R^2:x^3 =y^3\}$$ I know the equivalent definition but I can't explain why this should be a submanifold or not. Can please someone gives me some basic explanation, thank you!
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Hint:
The set is nothing but $\{(x,y)\in\mathbb{R}^2: y=x\}$.
Notes.
Also note that in order to talk about the notion of "submanifold", you should have some manifold in mind. Although, when the context is clear, people may not explicitly mention it.
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I guess using the inverse function theorem would be a good idea but I don‘t know how – Frederick Manfred Jan 19 '21 at 21:43
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@FrederickManfred nothing fancy here. This the graph of the function $f(x)=x$. – Jan 19 '21 at 21:45
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Okay, so we will have that $M$ is a submanifold, becasue it can be represented locally as graph (in fact also globally not?) – Frederick Manfred Jan 19 '21 at 21:48
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What is your definition of submanifolds? – Jan 19 '21 at 21:49
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We have 4 equivalent definitions, but one says: “$M$ is locally the graph of a smooth function. – Frederick Manfred Jan 19 '21 at 21:53
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@FrederickManfred: yes $M$ here is locally the graph of a smooth function. It is "globally" so in this example. – Jan 19 '21 at 21:54
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Hence it is a submanifold – Frederick Manfred Jan 20 '21 at 06:25