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Find all geodesic paths between neighboring peak points P1 and P2 on an egg crate surface of

$$ z= \sin x + \sin y, \; \text{ at}\ (x,y)=(\pi/2,\pi/2), (5 \pi/2), 5 \pi/2)$$

other than the obvious diagonal path connecting them ( through valley /trough bottom ) as intersection with $ x=y $ plane.

In 2D smooth surfaces there is one shortest among many locally geodesic paths(P1-P2)

In case ( think it is unlikely) this path is unique, how is it proved ?

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Narasimham
  • 42,260
  • I feel like you could argue there is no unique path (other than the diagonal path you excluded) due to the symmetry of the problem. Whatever path you find could be reflected along the diagonal and you would get the same result. – Michael M Dec 29 '20 at 12:05
  • The link shows multiplicity of admissible non minimum possible paths on a cylinder. – Narasimham Dec 29 '20 at 13:50

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