Suppose \(M_1,\ldots , M_k\) are manifolds with dimensions \(n_1,\ldots ,n_k\). The product \(M_1\times \cdots \times M_n\) is a \(n_1+\cdots +n_k\)-manifold.
Examples
Proof
The product satisfies as a product of second countable Hausdorff space the same properties (see (0x678f2168)
and (0x678f2213)
).
We check the local Euclidean property.
To construct a local chart for some point find local charts for every component of that point. The product of this charts is a homeomorphism onto an open subset of \(\mathbb{R}^{n_1+\cdots +n_k}\).