Continuity of Riemannian norm

Question

I just started learning something about riemannian manifolds.

I was wondering if the norm, as a map between the tangent bundle and $\mathbb{R}$ is continuous. Is it?

Thanks

Answer 1

The answer is yes.

By the fact that the transition maps are diffeomorphisms, if continuity holds in one chart, it will hold in every other chart as well. Thus it suffices to check continuity in some coordinate chart.

Let local coordinates of $TM$ be given by $(x_1,x_2,...,x_n,v^1,v^2,...,v^n)$. In those coordinates, the norm reads

$$\|v\|=\sqrt{g_{ij}(x_1,...,x_n)v^iv^j}.$$

Since $g$ is a Riemannian metric, the components $g_{ij}(x_1,...,x_n)$ are smooth in $(x_1,...,x_n)$, so in particular they are continuous. The contraction with the vector components is a sum of products of continuous functions, which produces again a continuous function which has nonnegative values, and the square root is also continuous on $[0,\infty)$. Hence, the norm is continuous in this coordinate chart.