Let p and q be distinct points of S2 If p notequalto q prove

Let p and q be distinct points of S^2. If p notequalto -q prove that the short great circle are is the unique shortest length joining p and q.

Solution

If p and q are distinct points of the sphere Sⁿ, and if C:[0,1]Sⁿ is a \"shortest path\" joining p to q,also
if F:SⁿSⁿ is a distance-preserving map fixing p and q, then FC is the shortest path (because the length of
FC is equal to the length of C).
Assume qp.If there exists a unique shortest path from p to q, then we can see
that the \"short\" great circle arc is the only arc because Every point not on the great circle through p
and q is moved by some isometry of the sphere that fixes p and q.
If we think specifically of S², then reflection F in the plane containing p,q, and the center of the sphere
is an isometry, and f(x)=x if and only if x lies on the great circle through p and q.