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unit tangent vector u , they have the same value when u is the tangent vector to g at
p . The only difference is that, as we have shown, k n,g is completely defined by the
tangent vector to g at p and thus basically a function defined on all tangent vectors,
whereas k n is only defined on unit tangent vectors. For that reason one can use the
symbols k n,g and k n interchangeably when unit tangent vectors are involved.
Using Lemma 9.9.5 and Theorem 1.8.10 one can show that T p ( S ) has an ortho-
normal basis ( u 1 , u 2 ) so that
( ( ) =-
D np u
k
u
1
1
1
(9.45)
( ( ) =-
D np u
k
u
.
2
2
2
Furthermore, if we assume that k 1 ≥ k 2 , then k 1 and k 2 are the maximum and minimum
of the second fundamental form Q II , respectively, when restricted to the unit circle of
the tangent space at p . The numbers k 1 and k 2 are also the maximum and minimum,
respectively, of the values of the normal curvatures k n at p .
Definition. The numbers k 1 and k 2 are called the principal normal curvatures to S
at p . The unit vectors u 1 and u 2 are called the principal normal directions for S at p .
The principal normal directions will be unique at those places where D n is non-
singular. These definitions agree with those that were given at the beginning of this
section. Furthermore, we can now easily prove Euler's formula (9.41) for the normal
curvature in any direction. Let u be a unit vector in T p ( S ). Then using the orthonor-
mal basis ( u 1 , u 2 ) used in Equation (9.45) we can express u in the form
= (
)
+ (
)
u
cos
q
u
sin
q
u
1
2
for some angle q. It follows that
()
k
=
Q
D
D
u
np u
n
II
( ()
=-
u
()(
(
)
+ (
)
) (
(
)
+ (
)
)
=-
n p
cos
q
u
sin
q
u
cos
q
u
sin
q
u
1
2
1
2
= (
(
)
+ (
)
) (
(
)
+ (
)
)
k
cos
q
u
k
sin
q
u
cos
q
u
sin
q
u
1
1
2
2
1
2
2
2
=
k
cos
q
+
k
sin
q
.
1
2
Definition. Let S be a surface and let D n ( p ):T p ( S ) Æ T p ( S ) be the Weingarten
map at a point p in S . The determinant of D n ( p ) is called the Gauss curvature of
S at p and is denoted by K( p ). The mean curvature of S at p , denoted by H( p ), is
derived from the trace of D n ( p ) and defined by
() =- ()
(
()
)
H
p
1 2 tr
D
n p
.
9.9.8. Theorem.
The two definitions of Gauss curvature agree.
Proof.
See [DoCa76] or [Spiv70b].
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