Product manifold - Knowlpedia

Let $M$ and $N$ be smooth manifolds of dimensions $m$ and $n$ . Their product manifold is the Cartesian product $M\times N$ , with the product topology and the unique smooth structure for which every pair of charts $(U,\varphi)$ on $M$ and $(V,\psi)$ on $N$ gives a chart

U\times V\longrightarrow \varphi(U)\times\psi(V),\qquad (p,q)\longmapsto(\varphi(p),\psi(q)).

It has dimension $m+n$ . The coordinate projections $\operatorname{pr}_M:M\times N\to M$ and $\operatorname{pr}_N:M\times N\to N$ are smooth.

Universal property

For any smooth manifold $P$ , a map $h:P\to M\times N$ is smooth exactly when both component maps $\operatorname{pr}_M\circ h$ and $\operatorname{pr}_N\circ h$ are smooth. Consequently $M\times N$ , together with its projections, is a categorical product in the category of smooth manifolds and smooth maps. This property determines the product up to a unique diffeomorphism compatible with the projections.

Tangent spaces and maps

There is a canonical vector-space isomorphism between tangent spaces:

T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN.

Under this identification, the differential of the first projection is the projection onto $T_pM$ , and similarly for the second. If $f:P\to M$ and $g:P\to N$ are smooth, the paired map $(f,g):P\to M\times N$ has differential $d(f,g)_x=(df_x,dg_x)$ . These facts follow directly from product coordinates.

Examples and scope

Euclidean space satisfies $\mathbb R^m\times\mathbb R^n\cong\mathbb R^{m+n}$ with its standard smooth structure. A finite product of smooth manifolds is again a smooth manifold, by iteration. Infinite Cartesian products generally are not finite-dimensional manifolds and are outside this definition. If manifolds with boundary are allowed, a product of two such manifolds naturally has corners rather than merely boundary, so it belongs to a larger category unless one factor has empty boundary.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. Publisher record. Relevant: Chapter 3, product manifolds and smooth maps.
Loring W. Tu, An Introduction to Manifolds, 2nd ed., Universitext, Springer, 2011. Publisher record. Relevant: Chapter 5, products and smooth structures.