Length (Curve)

Let $\gamma(t)$ be a smooth curve in a Manifold $M$ from $x$ to $y$ with $\gamma(0)=x$ and $\gamma(1)=y$. Then $\gamma'(t)\in T_{\gamma(t)},$ where $T_x$ is the Tangent Space of $M$ at $x$. The length of $\gamma$ with respect to the Riemannian structure is given by

$$\int_0^1 \vert\vert\gamma'(t)\vert\vert _{\gamma(t)}\,dt.$$

See also Arc Length, Distance