The first and second fundamental form in a surface are symmetric bilinear forms, in particular they are metrics. Of course, the first fundamental form is positive definite because is the Euclidean metric in the tangent plane. We consider the second fundamental form σp:TpS×TpS→R σp(v,v)=⟨−dNp(v),v⟩.
If X=X(u,v) is a parametrization of the surface, the matrix of σp with respect to the basis {Xu,Xv} is (effg).
We study the type of σp. However, it is better to choose a more suitable basis of TpS, indeed, a basis of principal directions because in such a case, σp→(κ1(p)00κ2(p)). Because this basis diagonalizes the metric σp, we conclude:
- The point is elliptic (K(p)>0) is equivalent to κi(p)>0 for i=1,2, or κi(p)<0 for i=1,2 and this means that σp is definite.
- The point is hyperbolic (K(p)<0) is equivalent to κ1(p)<0<κ2(p). Then σp is a non-degenerate indefinite metric with signature (1,1).
- K(p)=0, that is, some principal curvature is 0. We have two subcases.
- If the point is parabolic, then the non-zero principal curvature is positive or negative. This is equivalent to say that σp is positive semidefinite or negative semidefinite and also, the metric is degenerate. Here the radical of σp is the vector subspace spanned by the principal direction of the zero principal curvature.
- If the point is flat, then the metric is null.
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