Page 370 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY
P. 370
Unit 30: Geodesic Curvature and Christoffel Symbols
Introduction Notes
In this unit, we focus exclusively on the study of Geoderic Curvature. In this unit, we will go
through the properties of the curvature of curves on a surface. The study of the normal and of the
tangential components of the curvature will lead to the normal curvature and to the geodesic
curvature. We will study the normal curvature, and this will lead us to principal curvatures,
principal directions, the Gaussian curvature, and the mean curvature. In turn, the desire to
express the geodesic curvature in terms of the first fundamental form alone will lead to the
Christoffel symbols. The study of the variation of the normal at a point will lead to the Gauss
map and its derivative, and to the Weingarten equations.
30.1 Geodesic Curvature and the Christoffel Symbols
We showed that the tangential part of the curvature of a curve C on a surface is of the form k n .
g
g
We now show that k can be computed only in terms of the first fundamental form of X, a result
n
first proved by Ossian Bonnet circa 1848.
The computation is a bit involved, and it will lead us to the Christoffel symbols, introduced in
1869.
Since n is in the tangent space T (X), and since (X , X ) is a basis of T (X), we can write
g
p
u
v
p
k g g n = AX + BX ,
u
v
form some A, B .
However,
kn = k N + k g g n ,
N
and since N is normal to the tangent space,
.
.
N X = N X = 0, and by dotting
u
v
k g g n = AX + BX v
u
with X and X , since E = X u . X , F = X u . X , and G = X v . X , we get the equations:
v
v
u
v
u
.
kn X = EA + FB,
u
.
kn X = FA + GB.
v
On the other hand,
kn = X = X u + X v + X (u) + 2X uv + X (v) .
2
2
uv
uu
u
vv
v
Dotting with X and X , we get
v
u
.
kn X = Eu + Fv + (X uu . X )(u) + 2(X uv . X )uv + (X vv . X )(v) ,
2
2
u
u
u
u
kn . Xv = Fu + Gv + (X uu . X )(u) + 2(X uv . X )uv + (X vv . X )(v) .
2
2
v
v
v
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