Local coordinates on a product of two manifolds.

Question

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. I think that a local coordinate on $X \times Y$ is $(U \times V, x_1 \otimes y_1, \ldots, x_n \otimes y_m)$. Is this correct? Thank ou very much.

Answer 1

I'll do the case $n=m=1$ for ease of notation (nevermind the fact that all manifolds of dimension one are $S^1$...)

Let $X$ and $Y$ be the manifolds, and let $(U,x)$ be a local coordinate on $X$, and $(V,y)$ a local coordinate on $Y$ (this is by definition a homeomorphism $V \to \mathbb R$).

Then the charts on $X \times Y$ are given by all combinations of $x\times y: U \times V \to \mathbb R \times \mathbb R$. The map is $(a,b) \mapsto \left( x(a),y(b) \right)$.