Does the globally shortest path between two points on a surface in 3D satisfy the geodesic equation except for countably often?

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Let $P:[0,1]to S$ be the shortest path between two points on a compact regular surface. Will $P$ always be geodesic except for countably many turns, or could it be that there is a whole subinterval of $[0,1]$ where the path does not satisfy the geodesic equation?




geometry manifolds surfaces geodesic




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    I feel as though I am missing something - won't $P$ be geodesic everywhere?
    – Jason DeVito
    Aug 1 at 16:34














up vote
0
down vote

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Let $P:[0,1]to S$ be the shortest path between two points on a compact regular surface. Will $P$ always be geodesic except for countably many turns, or could it be that there is a whole subinterval of $[0,1]$ where the path does not satisfy the geodesic equation?




geometry manifolds surfaces geodesic




share|cite|improve this question















  • 2




    I feel as though I am missing something - won't $P$ be geodesic everywhere?
    – Jason DeVito
    Aug 1 at 16:34












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $P:[0,1]to S$ be the shortest path between two points on a compact regular surface. Will $P$ always be geodesic except for countably many turns, or could it be that there is a whole subinterval of $[0,1]$ where the path does not satisfy the geodesic equation?




geometry manifolds surfaces geodesic




share|cite|improve this question











Let $P:[0,1]to S$ be the shortest path between two points on a compact regular surface. Will $P$ always be geodesic except for countably many turns, or could it be that there is a whole subinterval of $[0,1]$ where the path does not satisfy the geodesic equation?





geometry manifolds surfaces geodesic





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 9:18









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  • 2




    I feel as though I am missing something - won't $P$ be geodesic everywhere?
    – Jason DeVito
    Aug 1 at 16:34












  • 2




    I feel as though I am missing something - won't $P$ be geodesic everywhere?
    – Jason DeVito
    Aug 1 at 16:34







2




2




I feel as though I am missing something - won't $P$ be geodesic everywhere?
– Jason DeVito
Aug 1 at 16:34




I feel as though I am missing something - won't $P$ be geodesic everywhere?
– Jason DeVito
Aug 1 at 16:34















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