Riemannian distance

Remarks

Example: on $\R^n$ with the Euclidean metric $g = \sum_i dx_i^2$, the straight segment from $p$ to $q$ has length $\abs{q - p}$, and no curve is shorter, so $d_g(p, q) = \abs{q - p}$.

Non-example: if $M$ is not connected, two points in different components are joined by no curve. Then the infimum is over the empty set, and $d_g$ is not a finite distance. Some authors then set $d_g(p, q) = \infty$.

The function $d_g$ is a metric on $M$, and the topology it induces is the given topology of $M$. Positivity $d_g(p, q) > 0$ for $p \ne q$ is the nontrivial part; it follows by comparing $g$ with a Euclidean metric in a chart. The infimum need not be attained: in $\R^2 \setminus \set{0}$ with the Euclidean metric, $d_g((-1, 0), (1, 0)) = 2$, but every curve between these points avoiding $0$ is longer than $2$.