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Equations (9.53) complete our analysis of D n and we are ready to draw some
important consequences. Incidentally, equations (9.48) with the values of c ij as shown
in (9.53) are called the Weingarten equations .
9.9.15. Theorem.
The Gauss curvature K satisfies
2
2
LN
-
-
M
LN
-
¥
M
K
=
=
.
2
2
FF
EG
F
u
v
Proof. The theorem follows from equation (9.52) and the definition of Gauss
curvature as the determinant of matrix (9.50). We see that K is just the quotient
of the discriminants of the first and second fundamental forms.
9.9.16. Theorem.
The mean curvature H satisfies
EN
+-
-
GL
2
FM
H
=
.
(
)
2
2
EG
F
Proof. The theorem follows from the definition of H and the values in (9.53) from
which we can immediately compute the trace of the matrix (9.50) for D n .
Finally, we relate the Gauss and mean curvatures to the principal normal curva-
tures k 1 and k 2 . Since the -k 1 and -k 2 are eigenvalues of D n , the linear map D n + kI
is not invertible when k = k 1 or k 2 . Therefore, its matrix has zero determinant, that is,
ckc
c
+
Ê
Ë
ˆ
¯
11
12
det
=
0
.
c
+
k
21
22
This expands to
2
(
)
(
) =
k
++
c
c k cc
+
-
cc
0
,
11
22
11
22
12
21
or
2
k
-
2
kK
+
=
0
,
(9.54)
for k = k 1 or k 2 . In other words, we have proved
9.9.17. Theorem. The principal normal curvatures satisfy Equation (9.54) and are
given by the formula
2
HHK
±
-
.
(9.55)
It is again time for an example. Analyzing the geometry invariably involves com-
puting the functions that are the coefficients of the first and second fundamental
forms. The functions L, M, and N are the more complicated. They were defined in the
derivation of equation (9.47). Each had basically two definitions. However, given the
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