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let $x_1$, $y_1$, $x_2$, $y_2$ be points on the sphere $S^n$ such that $x_1$ and $y_1$ are not antipodal and similarly for $x_2$, $y_2$. Why $x_1$ and $y_1$ can be joined by a path that do not cross with a path joining $x_2$ to $y_2$? and why we call these paths "shortest geodesics"?

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I think there must be some implicit assumptions missing in your question. You can join any two pairs of points on the sphere by paths that don't cross; that doesn't depend on the pairs not being antipodal. On the other hand, if you require the paths to be shortest geodesics, then these may cross even if the pairs are not antipodal. Please check your question carefully. – joriki Jul 16 '11 at 14:09

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If two points on a sphere are not identical and not antipodal then they lie on a unique great circle. The shortest distance between the two points will the shorter of the two arcs of the great circle between the two points

Two distinct great circles intersect twice. The two shorter arcs between the pair of points may or may not intersect. This diagram may help.

great circles

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You have assumed $n=2$. The two great circles $t\mapsto \cos t \ e_1+\sin t\ e_2$ and $t\mapsto \cos t\ e_3+\sin t\ e_4$ on $S^3$ do not intersect. – Christian Blatter Jul 16 '11 at 18:55

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