G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, using the metric on each $g_i(U_i)$s (the standard metric from the reals)? I tried defining some metric which for elements $x,y$ in two different $U_i$s makes $d(x,y) = inf_{t \in U_i \cap U_j} d_i(x,t) + d_j(t,y)$ where for $x,y \in U_i$ we set $d_i(x,y) = |g_i(x) - g_i(y)|$, but I can't manage to get the exact right definition which makes it a metric.
Thanks!