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Suppose I have a Geodesically complete Riemannian manifold $(M,g)$ with bounded curvature. Let $A,B,C$ be points on the $M$ and let $E_A^B,E_B^C,E_C^A$ be geodesic segments connecting $A,B,C$ respectively.

If I know the length of $E_A^B$ and of $E_B^C$ then can I explicitly compute $E_C^A$?

AB_IM
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    what do you mean by explicitly? which kinds of dependencies are allowed and which are forbidden? – Max Sep 08 '17 at 20:14
  • Is there a general closed-form formula or closed-form upper bound (finite :P ) – AB_IM Sep 08 '17 at 20:16
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    There is certainly no closed-form formula. If you are requiring the segments to be minimizing then $|E_A^B| +| E_B^C|$ is obviously an upper bound; and you can't really expect to do better than this without extra information. (That is, unless your "bounded curvature" includes a positive lower bound, in which case Myer's theorem could help.) – Anthony Carapetis Sep 09 '17 at 01:25
  • No its a negative lower bound and the upper bound is 0. – AB_IM Sep 12 '17 at 21:18

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