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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 7 '12 at 13:47
    
I posted a proof at: math.stackexchange.com/questions/141940/… – Ayman Hourieh May 27 '12 at 19:50

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