> [!definition] > > Let $X, Y$ be [[Manifold|manifolds]], $(U_i, \psi_i)_{i \in I}$, and $(V_j, \varphi_j)_{j \in J}$ be [[Atlas|atlases]] for $X$ and $Y$ respectively, then $(U_i \times V_j, \psi_i \times \varphi_j)_{(i, j) \in I \times J}$ is an atlas for $X \times Y$. If two pairs of atlases are [[Compatible Atlases|compatible]], then their products are also compatible. The manifold structure on $X \times Y$ is the **product** manifold. > > *Proof*. Since $\seqi{U}$ covers $X$ and $\seqj{V}$ covers $Y$, $\bracs{U_i \times V_j}_{(i, j) \in I \times J}$ covers $X \times Y$. As each map is a bijection, their product is also a bijection. > > Moreover, for any $i_1, i_2 \in I$, $j_1, j_2 \in J$, > $ > \begin{align*} > &\psi_{i_1} \times \varphi_{i_1}(U_{i_1} \times V_{j_1} \cap U_{i_2} \times V_{j_2}) \\ > &=\psi_{i_1} \times \varphi_{i_1}((U_{i_1} \cap U_{i_2}) \times (V_{j_1}\cap V_{j_2})) \\ > &= \psi_{i_1}(U_{i_1} \cap U_{i_2}) \times \varphi_{j_1}(U_{j_1} \cap U_{j_2}) > \end{align*} > $ > where both sets are open, so their product is open. > > Lastly, since each $\psi_i$ and $\varphi_j$ is a [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions)-isomorphism, their product is also a $C^p$-isomorphism. Therefore for suitable domains, > $ > (\psi_{i_1} \times \varphi_{j_1}) \circ (\psi_{i_2} \times \varphi_{j_2})^{-1} = (\psi_{i_1} \circ \psi_{i_2}^{-1}) \times (\varphi_{j_1} \circ \varphi_{j_2}^{-1}) > $ > is still a $C^p$-isomorphism. > > Now let $(U_i \times V_j, \psi_i \times \varphi_j)_{(i, j) \in I \times J}$ and $(S_k \times T_l, \sigma_k \times \tau_l)_{(k, l) \in K \times L}$ be atlases where $(U_i, \psi_i)_{i \in I}$ and $(S_k, \sigma_k)_{k \in K}$ are compatible, and $(V_j, \varphi_j)_{j \in J}$ and $(T_l, \tau_l)_{l \in L}$ are compatible. Then for suitable domains, > $ > (\psi_{i} \times \varphi_{j}) \circ (\sigma_{k} \times \tau_l)^{-1} = (\varphi_i \circ \sigma_{k}^{-1}) \times (\psi_{j} \circ \tau_{l}^{-1}) > $ > is still a $C^p$ isomorphism.