Suppose $G$ is a Lie group. Lemma 2.18 of Lectures on Lie Groups by J. Frank Adams states the following:
Let $\phi_\alpha : U_\alpha \to G_\alpha$ be a chart on G which sends $0 \in V$ to $e \in G$. Then, omitting $\phi_\alpha$, we can write $xy = x + y + o(r)$ in a neighbourhood of $e$ in $G$, where $r = r(x, y)$ denotes the distance of $(x, y)$ from $(e, e)$ in $G \times G$ under a metric.
This seems to be a Taylor expansion of the product around the origin. However, I am not used to such an expansion where the error term is given in terms of a metric on the manifold. How does this work?
In this particular example you can take any metric in a neighbourhood of $e$ – in particular, you can take the metric induced from $U_\alpha$ (which has a natural Euclidean metric) by your coordinate chart, and then literally have $r(x,y) = \sqrt{\lVert x\rVert^2+\lVert y\rVert^2}$.
More generally, any Riemannian metric on a manifold $M$ induces a distance function such that $d(m,\cdot)$ in a neighbourhood of any point $m\in M$ is equivalent (up to some multiplicative constants) to $\left(\sum_{i=1}^n x_i^2\right)^{1/2}$ in any coordinate chart which sends $m$ to $0$. (Sketch of proof: take local coordinates in which the Riemannian metric at $m$ is diagonal and use the fact that the Riemannian metric is smooth, so it varies not too much in the neighbourhood of $m$).