Let $M$ and $N$ be manifolds and let $q_0$ be a point in $N$. Prove that the inclusion map $i_{q_0} : M \to M×N : p \mapsto (p,q_0)$, is $C^\infty$.
The smoothness of an inclusion map
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manifolds
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0Let M and N be manifolds and let q0 be a point in N. Prove that the inclusion map iq0 : M → M×N, iq0 (p) = (p,q0), is c^∞. – 2012-10-24
1 Answers
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If we assume that $M$ and $N$ are smooth manifolds, try this:
Let $p\in U\subset M$ and let $(\phi,U)$ be a chart on $M$. Also, let $(p,q_0)\in U\times W\subset M\times N$ and let $(\psi,U\times W)$ be a chart on $M\times N$. Then the problem reduces to smoothness of the induced map $\tilde{i_{q_0}}:\phi(U)\rightarrow \psi|_{U\times \{q_0\}}(U)$ where $x\mapsto (x,y_0)$ for a fixed $y_0$, which we know is smooth.
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0@hamid Are you a bot or a real perso$n$? – 2012-11-02