Are geodesics locally minimizing in continuous curves? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:56:29Z http://mathoverflow.net/feeds/question/68921 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68921/are-geodesics-locally-minimizing-in-continuous-curves Are geodesics locally minimizing in continuous curves? Sebastian Scholtes 2011-06-27T12:36:04Z 2011-06-27T16:35:27Z <p>In every lecture on Riemannian geometry it is standard to prove that geodesic curves are locally length minimizing. The only thing I find confusing about this is, that here length minimizing means: compared to all piecewise smooth curves in contrast to, say, all continuous curves. So my question is:</p> <p>Are geodesics locally length minimizing in the continuous curves?</p> <p>If generally they are not: Under which conditions can we obtain such a result? Can you give any counterexamples?</p> http://mathoverflow.net/questions/68921/are-geodesics-locally-minimizing-in-continuous-curves/68945#68945 Answer by Anton Petrunin for Are geodesics locally minimizing in continuous curves? Anton Petrunin 2011-06-27T16:35:27Z 2011-06-27T16:35:27Z <p>Your question will be trivial once you give a definition of the length of curve in a Riemannian manifold.</p> <p>For example, you may define distance as infimum of lengths piecewise smooth curves connecting given points. Then you define length of general curve as you do it in a metric space...</p>