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.
Geodesics are in fact the paths of zero (intrinsic) curvature. The standard definition of a geodesic requires that the covariant derivative of its unit tangent vector along the curve be $0$.