How can it be shown that the only such surfaces are spheres or planes?
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This is exactly Proposition 4 in section 3-2 in Do Carmo's "Differential Geometry of curves and surfaces", P.147. You can find the full proof there. – Paul Mar 07 '12 at 13:47
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I posted a proof at: http://math.stackexchange.com/questions/141940/umbilic-points-on-a-connected-smooth-surface-problem/141945#141945 – Ayman Hourieh May 27 '12 at 19:50