let $(M,g)$ be a Riemannian Manifold, let $ \gamma : [a,b] \rightarrow M$ be a piecewise smooth curve and let $\Omega : M \rightarrow \mathbb{R}^{n}$ be a coordinate chart. The length of $\gamma$ on the manifold is biven by

$$L_{g}(\gamma) = \int_{a}^{b} | \gamma'(t)|_{g} dt$$.

As with everything in differential geometry, the hard part is in the interpretation.

$\gamma'(t)$ is an element of the tangent space to M at $\gamma(t)$. In local coordinates it is given by the equation 

$$\gamma'(t) = (\gamma^{i})'(t) \frac{\partial}{\partial x^{i}} | _{\gamma(t)}$$ 

where $\gamma^{i}$ is composed with the coordinate chart composed with the ith projection map. By definition

$$| \gamma'(t)|_{g} = (g_{\gamma(t)}(\gamma'(t),\gamma'(t)))^{\frac{1}{2}}$$

Edit:still editing!