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