Is a geodesic the least curved path?

Is a geodesic the least curved path?

It is clear that, in $\mathbb{R}^n$, straight lines are the lines with
minimum possible curvature. That is, given the Frenet-Serret
($n$-dimensional equivalent) matrix, and taking its squared norm, we can
integrate it along the line and get zero on a straight line, and a
positive number on any other curve.
Can we generalize this idea to arbitrary Riemannian manifolds? (What about
any affine connection?)
In other words. Is there a variational principle on curvature, rather than
on length?
Any reference is welcome.