Is $f:X\rightarrow Y$ smooth if and only if the graph $\Gamma_f$ is a closed submanifold of $X\times Y$? Thank you very much.
2026-04-04 07:54:11.1775289251
Quick question: Map being smooth vs Graph being submanifold of the product space
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I guess not. Let $f:\mathbb R\to\mathbb R$ be $f(x)=x^{1/3}$. The graph is a closed submanifold of $\mathbb R^2$. However the map is not smooth.