Does a geodesic locally minimize the curvature?

Question

Does a geodesic somehow minimize the local curvature? Lets say we have a smooth geodesic $\gamma$ with constant speed on a surface $S\subset\mathbb{R}^n$ and with $\gamma(t)=x$, then every other smooth curve $\eta$ on $S$ with the same speed such that $\eta(t)=x$ has a a curvature which is greater or equal to the one of $\gamma$ at $t$? Does this somehow characterizes the geodesic?