Page 373 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes and
g = [ ; ] + [ ; ].
|
From all this, we get
2[ ; ] = g + g – g .
| | |
As before, the Christoffel symbols [ ; ] and  are related via the Riemannian metric by the


equations
 g 11 g 12   1
      [ ; ].

 g 21 g 22 
This seemingly bizarre approach has the advantage to generalize to Riemannian manifolds.
30.2 Principal Curvatures, Gaussian Curvature, Mean Curvature

We will now study how the normal curvature at a point varies when a unit tangent vector varies.
In general, we will see that the normal curvature has a maximum value k and a minimum value
1
k , and that the corresponding directions are orthogonal. This was shown by Euler in 1760.
2
The quantity K = k k called the Gaussian curvature and the quantity H = (k + k )/2 called the
1
2
1 2
mean curvature, play a very important role in the theory of surfaces.
We will compute H and K in terms of the first and the second fundamental form. We also classify
points on a surface according to the value and sign of the Gaussian curvature.
Recall that given a surface X and some point p on X, the vectors X , X form a basis of the tangent
u
v
space T (X).
p

Given a unit vector t = X x + X y, the normal curvature is given by
u
v

k ( t ) = Lx + 2Mxy + Ny ,
2
2
N
since Ex + 2Fxy + Gy = 1.
2
2
Usually, (X ,X ) is not an orthonormal frame, and it is useful to replace the frame (X ,X ) with an
v
v
u
u
orthonormal frame.
 
One verifies easily that the frame (e ,e ) defined such that
2
1
 X  EX  FX
1 e  u , 2 e  v u .
E E(EG F )
2

is indeed an orthonormal frame.
  
With respect to this frame, every unit vector can be written as t  cos 1 e  sin  2 e , and expressing
 
(e 1,e ) in terms of X and X , we have
2
u
v
  w cos  Fsin  E sin
t    X  X ,
v
u
 w E  w
2
where w = EG F .

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