I'm new to Riemannian geometry, trying to figure out if it's the tool I need before diving in head first. Sorry if this is a basic question.
Say I have a smooth manifold $M$ of dimension $n$ with a chart $\phi$. Additionally, let $C$ be a family of smooth curves, $c \in C: [0 , 1] \rightarrow \mathbf{R}^{n}$. So for all $c \in C$, $\phi^{-1}\circ c$ is a smooth curve on the manifold. See the edit below.
If I have a way to naively assign a non-negative "length" to $\phi^{-1}\circ c$ for all $c \in C$, are there sufficient conditions for whether M can be given a Riemannian structure that assigns each $\phi^{-1}\circ c$ this length?
Alternatively, a resource that discusses this would be just as useful. Thank you!
Edit: In all of the cases that I'm considering, the family of curves $C$ is homeomorphic to $\mathbf{R}^{m}$ for some $m$. For example, the family of circles parameterized by the two coordinates of their center and by their radius, inheriting the topology of its parameter space. With this topology, we can assume that the "length" function $l: C \rightarrow \mathbf{R}$ is continuous.
