Let be a surface element. Suppose that the image of lies in the region , and the principal curvatures of at satisfy . Show the tangent plane of at that point is not parallel to the axis.

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- October 6th 2009, 02:50 PMchiph588@Hyperbolic point
Let be a surface element. Suppose that the image of lies in the region , and the principal curvatures of at satisfy . Show the tangent plane of at that point is not parallel to the axis.

- December 29th 2010, 08:56 AMRebesques

Clearly f(0)=0 is minimum and unique.

We use Taylor's expansion in a neighbourhood of 0, to obtain

or

with the obvious notation.

We also have . One of the principal curvatures at 0 must be negative, say . Since this is an eigenvalue of

there is a direction on the plane, of length , such that . Along this direction, we will have

a smooth function.

Therefore, there is an such that , a contradiction.