Geodesic equals energy minimum

Statement

Let $M$ be a complete Riemannian manifold and $p,q\in M$ be any two points in $M$ at distance $d$. Then, the energy functional on the path space $\Omega (M;p,q)$ attains its minimum value $d^{2}$ precisely on the minimal geodesics from $p$ to </math>q</math>.