Blaga P. Lectures on the differential geometry of - tiera.ru

4.16. Principal directions and curvatures 161
4.16.2 The computation of the curvatures of a surface
Theorem. Let r : U → R 3 be a local parameterization of an oriented surface S . Then
the total and the mean curvatures of S are given, respectively, by the formulas:
Kt = DD′′ − D ′2
H 2
(4.16.1)
Km = DG − 2D′ F + D ′′ E
2H2 . (4.16.2)
Proof. As we saw earlier, the matrix of the shape operator of a surface is given by
A = 1
H2 �
GL − FM
−FL + EM
�
GM − FN
−FM + EN
or A = 1
H2 �
FD ′ − GD
FD − ED
FD ′′ − GD ′
′ FD ′ − ED ′′
�
,
therefore,
Kt = det A = 1
H4 �
2 ′2 ′ ′′ ′ ′′ 2 ′′ ′ ′′
F D − EFD D − FGDD + EGDD − F DD + EFD D +
⎡
⎤
+FGDD ′ − EGD ′2� = 1
H 4
Km = − 1 1
TrA = −
2 2H2 ⎢⎣ (EG − F2 ) ·(DD
����������������
′′ − D ′2 )
⎥⎦ = DD′′ − D ′2
,
H 2
H 2
� ′ ′′
2FD − GD − ED � = DG − 2D′ F + D ′′ E
2H2 .
Corollary. The principal curvatures k1 and k2 are the roots of the equation
i.e.
Corollary. A non-flat point of a surface is
1. elliptic iff Kt > 0;
2. parabolic iff Kt = 0;
3. hyperbolic iff Kt < 0.
k 2 − 2Km · k + Kt = 0, (4.16.3)
�
k1 = Km + K2 m − Kt,
�
(4.16.4)
k2 = Km − K2 m − Kt. (4.16.5)
�