# The smoothness of an inclusion map

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 $C^\infty$.

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What is C¥? Is it $C^{\infty}$? –  Tomás Oct 24 '12 at 19:28
Let 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^∞. –  Ciris Oct 24 '12 at 19:42

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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whay Let p∈U⊂M and let (ϕ,U) be a chart on M . Also, let (p,q 0 )∈U×W⊂M×N and let (ψ,U×W) be a chart on M×N . Then the problem reduces to smoothness of the induced map i q 0 ~ :ϕ(U)→ψ| U×{q 0 } (U) where x↦(x,y 0 ) for a fixed y 0 , which we know is smooth. –  Ciris Oct 29 '12 at 17:10
@hamid Are you a bot or a real person? –  Matt N. Nov 2 '12 at 13:01