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The "monkey saddle" is a parametric surface defined by

$$ \begin{eqnarray} x & = & u \\ y & = & v \\ z & = & u^3 - 3 v^2 u \end{eqnarray} $$

Its second fundamental form has the coefficients $e = f = g = 0$ at the origin $(0, 0, 0$).

The Dupin indicatrix of a surface at some point $p$ is the set of tangent vectors $v$ for which $II_p(v) = 1$.

Here is a picture of the indicatrix of the monkey saddle at $(0, 0, 0)$:

enter image description here

[Taken from Dirk Jan Struik, Lectures on classical differential geometry (bottom of page 85).]

Why does the indicatrix of the monkey saddle exist at all at (0, 0, 0)? Since $e = f = g = 0$, we get $II(v) = 0 \neq 1$ for all $v$, so shouldn't it be the empty set?

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The original definition of the Dupin indicatrix is obtained by taking a plane that is parallel to, and small distance away from the tangent plane at a point, and examining the intersection of your surface with that plane. (See, for example, http://en.wikipedia.org/wiki/Dupin_indicatrix ; the same description is also made on the same page of the textbook you linked to.)

At points where the second fundamental form is non-vanishing, a Taylor-expansion process shows that to leading order the contribution to the shape of the indicatrix is precisely the second fundamental form. But at a point where the second fundamental form is vanishing, the "graphically constructed" Dupin indicatrix will be given by the next non-vanishing order in the Taylor-expansion.

The image for the Monkey Saddle that you linked to is drawn using the classical graphical method that Dupin defined. Now, noting that if the second fundamental form is non-vanishing, the "equation" solved by the indicatrix is

$$ \kappa_1 x^2 + \kappa_2 y^2 = 1 $$

which is either an ellipse, a hyperbola, or two parallel lines. That is, the picture must admit two reflection symmetries across the directions of principal curvatures, which intersect perpendicularly! The Monkey saddle, on the other hand, has a different symmetry! It has reflection symmetries across three different lines. This shows you that it, roughly speaking, has a vanishing second order Taylor coefficient (the second fundamental form; the first order is the tangent plane, and the zeroth order is the point itself), but a non-vanishing third order coefficient. (This is also more obvious if you use complex coordinates on the $x$-$y$ plane and consider the Monkey Saddle as the graph of $z = \Re (x+iy)^3$.)

So the image in the text, I imagine, is meant to illustrate the following facts:

  • The "graphical" construction of Dupin indicatrix is non-vanishing unless the surface is "flat to infinite order" (all Taylor coefficients vanishes).
  • If the second fundamental form is non-vanishing, the indicatrix is given by the "level set" of the second fundamental form; but if the second fundamental form is vanishing, the Dupin indicatrix can be non-vanishing and is given by the next order expansion in the Taylor polynomial.
  • The symmetries of the Dupin indicatrix tells you to which order the Taylor polynomial vanish.
  • In particular, if around a point $p$ in the manifold, the manifold exhibits locally an order $k > 2$ symmetry, the second fundamental form must vanish at the point.

And when it mentioned that the indicatrix for the monkey saddle is "constructed according to the preceding rule" on page 85 of the textbook you linked to, the rule it refers to is the original graphical construction of Dupin.

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Thank you... that makes sense! –  koletenbert Sep 3 '11 at 16:34

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