Why is it a submanifold or not?

Question

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!

Answer 1

Hint:

The set is nothing but $\{(x,y)\in\mathbb{R}^2: y=x\}$.

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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.