Geodesic equals energy minimum: Difference between revisions

Latest revision as of 19:41, 18 May 2008

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 $Ω (M; p, q)$ attains its minimum value $d^{2}$ precisely on the minimal geodesics from $p$ to $q$ .

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	==Statement==		==Statement==

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