Hyperbolic point: there are points at both sides of tangent plane

differential-formsdifferential-geometrysurfaces

I am studying a proof of the following facts, at the context of Gauss Aplication (Differential Geometry of Surfaces):

Given an elliptic point $p_0\in S$, there is a neighborhood of $p_0$ s.t. the points are all at same side of tangent plane $T_{p_0}S$.

In any neighborhood of a hyperbolic point $p_0\in S$, there are points at both sides of tangent plane $T_{p_0}S$.

To prove this, the author prove that for a paremetrization $\psi$ s.t. $\psi(0,0)=p_0$:

$d(u,v):=\langle \psi(u,v)-p_0,N(p_0)\rangle$ is s.t. $d(u,v)=\dfrac{1}{2}II_{p_0}(w)+r$, where $r$ is the rest of Taylor's formula, $II$ the second fundamental form and $w=u\psi_u+v\psi_v$.

OK until here, but I could not understand how states here that $II_{p_0}$ has same signal in the first case and alternates signal at the case hyperbolic.

Many thanks for the help.

Best Answer

Using the classical notation $$e=\langle \psi_{uu},N\rangle,\quad f=\langle \psi_{uv},N\rangle\quad\mbox{and}\quad g=\langle \psi_{vv},N\rangle,$$Sylvester's Criterion from Linear Algebra says that $${\rm II}_{p_0}=\begin{pmatrix} e(0,0)& f(0,0) \\ f(0,0)& g(0,0)\end{pmatrix}$$ is

  • positive-definite if $K(p_0)>0$ and $e(0,0)>0$;
  • negative-definite if $K(p_0)>0$ and $e(0,0)<0$;
  • indefinite if $K(p_0)<0$.

We are in the third case. So there are $(u_1,v_1)$ and $(u_2,v_2)$ such that $d(u_1,v_1)>0$ and $d(u_2,v_2)<0$.

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