I am reading a book where they define the Riemannian distance between two points on a manifold. Naturally it is given as the infimum of the integral of the length of the curves which connect the two points:
However, then they make a reparameterization of the curve, and suddenly the distance and the `norm' inside the integral gets squared. How does this happen? And also is the change of variables in the reparameterisation $s=\frac{t-a}{b-a}$?
There's an application of Jensen's inequality happening here, which is why the unit interval is used in the second definition.
The goal is to show that the two definitions of distance coincide. The integrals appearing in the first equation are reparameterization-invariant, so the first infimum can be taken to be over curves of constant speed on the unit interval. For such curves, $(\int_0^1 |\gamma'(t)| dt)^2 = \int_0^1|\gamma'(t)|^2 dt$, so that the second definition of distance is no larger than the first. For the other inequality, Jensen's inequality shows that for any curve $\gamma: [0,1] \to M$ with $\gamma(0) = x, \gamma(1) = y$, $$\int_0^1 |\gamma'(t)|^2 dt \geq \left( \int_0^1 |\gamma'(t)| dt \right)^2, $$ so that the two definitions of distance are the same.