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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.2012-03-07
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    I posted a proof at: http://math.stackexchange.com/questions/141940/umbilic-points-on-a-connected-smooth-surface-problem/141945#1419452012-05-27

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