Page 377 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Figure 30.2: Portion of Torus

The normal curvature
k (X x + X y) = Lx + 2Mxy + Ny 2
2
v
u
N
will vanish for some tangent vector (x, y)  (0, 0) iff M  LN  0.
2
Since
LN – M 2
K = ,
EG – F 2
this can only happen if K  0.
If L = N = 0, then there are two directions corresponding to X and X for which the normal
v
u
curvature is zero.
2
 x  x
If L  0 or N  0, say L  0 (the other case being similar), then the equation L    2M  N  0
y
  y
has two distinct roots iff K < 0.
The directions corresponding to the vectors X x + X y associated with these roots are called the
v
u
asymptotic directions at p. These are the directions for which the normal curvature is null at p.
There are surfaces of constant Gaussian curvature. For example, a cylinder or a cone is a surface
of Gaussian curvature K = 0. A sphere of radius R has positive constant Gaussian curvature
1
K = .
R 2

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