Atlas on a Product Manifold

Suppose \(M_1,\ldots , M_k\) are manifolds with dimensions \(n_1,\ldots ,n_k\), respectively. The local charts on the product manifold \(M_1\times \cdots \times M_k\) are of the form

\[ (U_1\times \cdots \times U_k, \varphi_1\times \cdots \varphi_k), \]

where, for \(i=1,\ldots ,k\), \((U_i,\varphi_i)\) is a local chart on \(M_i\).