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Prove or disprove ;

Let $S $ be a surface in $ R^3$. $S $ is a plane iff every point of $S $ is planar point.

"All points of plane are planar points" is trivial. But,... the converse is also really true?

The definition of planar point ; $p $ is called a planar point of $S $ iff the two principal curvatures vanish.

Chris kim
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  • The term "planar point" for a smooth surface in 3-space means a point where both principal curvatures equal zero. This contradicts the claim " is a plane iff every point of is planar point." – Dan Asimov Mar 07 '23 at 19:52

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Hint: The two principal curvatures are the directional derivatives of the normal vector to the surface in the principal directions. If they are both zero, what can you say about the normal vector?

treble
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  • The principal curvatures are the maximum and minimum of normal curvatures. So the every normal curvature is zero at any point in given surface. – Chris kim Aug 20 '13 at 02:44
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If the principal curvatures vanish on a connected surface I think you can show the shape operator vanishes identically. Then, see The Shape Operator on a path-connected open subset of a surface to complete the argument.

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James S. Cook
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