Page 394 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

The quantity K = k k called the Gaussian curvature and the quantity H = (k + k )/2 called Notes
1
1 2
2
the 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
v
u
tangent 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
Given a surface X, for any point p on X, letting A, B, H be defined as above, and

C = A + B , unless A = B = 0, the normal curvature k at p takes a maximum value k and
2
2
N 1
a minimum value k called principal curvatures at p, where k = H + C and k = H – C. The
1
2
2
directions of the corresponding unit vectors are called the principal directions at p.
It can be shown that a connected surface consisting only of umbilical points is contained in

a sphere.
It can also be shown that a connected surface consisting only of planar points is contained
in a plane.
A surface can contain at the same time elliptic points, parabolic points, and hyperbolic
points. This is the case of a torus.
The parabolic points are on two circles also contained in two tangent planes to the torus
(the two horizontal planes touching the top and the bottom of the torus on the following
picture).
The elliptic points are on the outside part of the torus (with normal facing outward),
delimited by the two parabolic circles.
The hyperbolic points are on the inside part of the torus (with normal facing inward).

30.10 Keywords

Christoffel symbols: The computation is a bit involved, and it will lead us to the Christoffel
symbols, introduced in 1869.
Gaussian curvature: The quantity K = k k called the Gaussian curvature and the quantity
1
2
H = (k + k )/2 called the mean curvature, play a very important role in the theory of surfaces.
1
2
Elliptic: At an elliptic point, both principal curvatures are non-null and have the same sign. For
example, most points on an ellipsoid are elliptic.
Hyperbolic: At a hyperbolic point, the principal curvatures have opposite signs. For example, all
points on the catenoid are hyperbolic.

Jacobian matrix: The Jacobian matrix of dN in the basis (X , X ) can be expressed simply in
p
u
v
terms of the matrices associated with the first and the second fundamental forms (which are
quadratic forms).

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