Nov 11, 2004
1
Curves and Surfaces
Fall 2004, PSU
Lecture Notes 12
2.6 Gauss’s formulas, and ChristoFel Symbols
Let
X
:
U
→
R
3
be a proper regular patch for a surface
M
, and set
X
i
:=
D
i
X
. Then
{
X
1
,X
2
,N
}
may be regarded as a
moving bases
of
frame
for
R
3
similar to the Frenet
Serret frames for curves. We should emphasize, however, two important
di±erences: (i) there is no canonical choice of a moving bases for a surface
or a piece of surface (
{
X
1
,X
2
,N
}
depends on the choice of the chart
X
);
(ii) in general it is not possible to choose a patch
X
so that
{
X
1
,X
2
,N
}
is
orthonormal (unless the Gaussian curvature of
M
vanishes everywhere).
The following equations, the ²rst of which is known as
Gauss’s formulas
,
may be regarded as the analog of FrenetSerret formulas for surfaces:
X
ij
=
2
X
k
=1
Γ
k
ij
X
k
+
l
ij
N,
and
N
i
=
−
2
X
j
=1
l
j
i
X
j
.
The coeﬃcients Γ
k
ij
are known as the
ChristoFel symbols
, and will be deter
mined below. Recall that
l
ij
are just the coeﬃcients of the second fundamen
tal form. To ²nd out what
l
j
i
are note that
−
l
ik
=
−h
N, X
ik
i
=
h
N
i
,X
k
i
=
−
2
X
j
=1
l
j
i
h
X
j
,X
k
i
=
−
2
X
j
=1
l
j
i
g
jk
.
Thus (
l
ij
)=(
l
j
i
)(
g
ij
)
.
So if we let (
g
ij
):=(
g
ij
)
−
1
,
then (
l
j
i
)=(
l
ij
)(
g
ij
)
,
which
yields
l
j
i
=
2
X
k
=1
l
ik
g
kj
.
1
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 Spring '08
 Staff
 Geometry, Formulas, ij, Riemannian geometry, lij lk Xl, Γp Γl

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