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If $\{(U_\alpha ,f_\alpha )\}$ and $\{(V_i,y_i)\}$ are $C^\infty$ atlases for the manifolds $M$ and $N$ of dimensions $m$ and $n$, respectively, then the collection {$(U_\alpha \times V_i,f_\alpha \times y_i : U_\alpha \times V_i →R^m \times R^n)$} of charts is a $C^\infty$ atlas on $M\times N$. Therefore, $M×N$ is a $C^\infty$ manifold of dimension $m+n$.


If $M$ and $N$ are $C^\infty$ manifolds, then $M×N$ with its product topology is Hausdorff and second countable.also locally Euclidean part is also easy.

But How can I able to show that the transition maps are $C^\infty$? Can Someone Explain in detail please. Thanks for your time.

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The transition map: $$ (f_\alpha\times y_i)\circ (f_\beta\times y_j)^{-1}=(f_\alpha\circ f_\beta^{-1})\times (y_i\circ y_j^{-1}) $$ Since $f_\alpha\circ f_\beta^{-1}$ and $y_i\circ y_j^{-1}$ are both smooth, the product is also smooth, since coordinates of $M$ and $N$ don't interact with each other (their partial derivatives should be consistently $0$, and therefore smooth). Explicitly, $$ (f_\alpha\circ f_\beta^{-1})\times (y_i\circ y_j^{-1}):(x^1,\dots,x^m;a^1,\dots,a^n)\mapsto(z^1,\dots,z^m;b^1,\dots,b^n) $$ From smoothness of $f_\alpha\circ f_\beta^{-1}$ and $y_i\circ y_j^{-1}$, $\partial z^i/\partial x^j$ and $\partial b^i/\partial a^j$ and higher order partial derivatives are all smooth, and the cross terms $\partial b^i/\partial x^j,\partial z^i/\partial a^j$ and the higher order partial derivatives are all $0$ and is therefore also smooth.

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