Page 374 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

 Notes
We can now compute K (t), and we get
N
2

  w cos  Fsin  2  (wcos  Fsin )sin  Esin 
k (t) = L    2M    N .
N
 w E   w 2  w 2
We leave as an exercise to show that the above expression can be written as

k (t) = H + Acos 2+ B sin 2,
N

where
GL – 2FM + EN
H = ,
2(EG – F )
2
L(EG – 2F ) + 2EFM – E N
2
2
A =
2E(EG – F )
2
EM – FL
B = .
E EG – F 2
2
2
Letting C = A  B , unless A = B = 0, the function
f() = H + Acos 2 + B sin 2
has a maximum k = H + C for the angles  and  + , and a minimum k = H – C for the angles
2
1
0
0
 3 A B
 + 2 and  + 2 , where cos 2 = C and sin 2 = C .
0
0
0
0
The curvatures k and k play a major role in surface theory.
2
1
Definition 1: Given a surface X, for any point p on X, letting A, B, H be defined as above, and C
2
2
= A + B , unless A = B = 0, the normal curvature k at p takes a maximum value k and a
1
N
minimum value k called principal curvatures at p, where k = H + C and k = H – C. The
2
1
2
directions of the corresponding unit vectors are called the principal directions at p.
k + k
The average H = 1 2 of the principal curvatures is called the mean curvature, and the
2
product K = k k of the principal curvatures is called the total curvature, or Gaussian curvature.
1 2
Observe that the principal directions  and  +  corresponding k and k are orthogonal.
2
0
2
1

2
2
2
2
2
Notes K = k k = (H – C)(H + C) = H – C = H – (A + B ).
1 2
After some laborious calculations, we get the following (famous) formulae for the mean curvature
and the Gaussian curvature:
GL – 2FM + EN
H = ,
2(EG – F )
2
LN – M 2
K = .
EG – F 2
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