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We add a bit to On partitioning the surface of a convex solid into geodesically convex equal area regions

  1. Consider a convex 3D solid body C - not necessarily a polyhedral body. What could be said about a C with the property: for any pair of points on its surface, the shortest path between them (geodesic) along the surface (if there are more than one shortest path, at least one of them) lies entirely in a plane? If C is a sphere, such a property holds because all geodesics are great circle arcs. But what other surfaces have this property?

  2. The sphere also has the property - for every pair of points on its surface, there is one unique geodesic (shortest path) between them, except when they are antipodal points when there are infinitely many such geodesics. Are there convex solids such that for every pair of points on the surface there are say, 2 different geodesics between them?

Note 1: Consider the surface of a cube. If we choose the centers of opposite faces as a pair of points, there are 4 different shortest geodesics between them. What we seek here is a surface such that every pair of points chosen on it will be joined by 2 different shortest paths.

Note 2: On a plane (which is not closed convex), every pair of points is joined by a unique shortest geodesic. I don't know if there is a convex solid whose surface has exactly 1 shortest geodesic joining any two pair of points on the surface.

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    $\begingroup$ If a surface is smooth (or merely $C^2$), having all its geodesics be planar is equivalent to the surface consisting entirely of umbilic points, i.e., the surface has to be part of a plane or a sphere. This is a simple consequence of the structure equations. $\endgroup$
    – Robert Bryant
    Commented Oct 10 at 9:37
  • $\begingroup$ Thank you. hope to have an answer to second query too $\endgroup$
    – Nandakumar R
    Commented Oct 10 at 14:08
  • $\begingroup$ I think you need to ask a more precise version of your 'second query'. The parenthetical remark at the end seems to be asking about a different question. $\endgroup$
    – Robert Bryant
    Commented Oct 10 at 22:52
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    $\begingroup$ So now that you've clarified Note 1, I can say that, if the surface is $C^2$, then no surface (compact or not) has the property that every pair of points can be joined by two distinct geodesics of shortest length. In fact, every point $p$ of such a surface $S$ has an open neighborhood $U$ such that any two points in $U$ have a unique shortest geodesic in $S$ joining them. (Of course, I don't count reparametrizations of a geodesic as different geodesics.). $\endgroup$
    – Robert Bryant
    Commented Oct 11 at 11:08
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    $\begingroup$ Moreover, for Note 2, if the solid is compact and its surface is $C^2$, then, for any point $p$ on the surface, there will always a point $q$ for which 'the' shortest geodesic joining $p$ to $q$ is not unique. You should look up 'cut locus' on Wikipedia to see what I mean. $\endgroup$
    – Robert Bryant
    Commented Oct 11 at 11:17

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