Page 382 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

This perpendicular distance can be expressed as Notes
.
(u, v) = (X(u, v) – X(u , v )) N (u ,v ) .
0
0
0 0
However, since X is at least C -continuous, by Taylor’s formula, in a neighborhood of (u , v ),
3
0
0
we can write
1
X(u, v) = X(u , v ) + X (u – u ) + X (v – v ) + 2 (X (u – u ) + 2X (u – u )(v – v ) + X (v – v ) ) +
2
2
0
u
0
uv
0
0
vv
0
uu
0
0
0
v
((u – u ) + (v – v ) )h (u, v),
2
2
0
1
0
where lim (u,v)  (u ,v ) h (u, v) = 0.
1
0 0
However, recall that X and X are really evaluated at (u , v ) (and so are X , X , and X ), and so,
v
uu
0
vv
u,v
0
u
they are orthogonal to N (u ,v ) .
0 0
From this, dotting with N (u ,v ) , we get
0 0
(u, v) = (L(u – u ) + 2M(u – u )(v – v ) + N(v – v ) ) + ((u – u ) + (v – v ) )h(u, v),
2
2
2
2
0
0
0
0
0
0
where lim (u,v)  (u ,v ) h(u, v) = 0.
0 0
Therefore, we get another interpretation of the second fundamental form as a way of measuring
the deviation from the tangent plane.
For  small enough, and in a neighborhood of (u , v ) small enough, the set of points X(u, v) on
0
0
1
2
the surface such that (u, v) =   will look like portions of the curves of equation
2
1 2 2 1 2 .
0
0
0
0
2 (L(u – u ) + 2M(u – u )(v – v ) + N(v – v ) )   
2
Letting u – u = x and v – v = y, these curves are defined by the equations
0
0
Lx + 2Mxy + Ny = ±1.
2
2
These curves are called the Dupin indicatrix.
 
It is more convenient to switch to an orthonormal basis where e and e are eigenvectors of the
2
1
Gauss map dN (u ,v ) .
0 0
If so, it is immediately seen that
Lx + 2Mxy + Ny = k x + k y , 2
2
2
2
2
1
where k and k are the principal curvatures. Thus, the equation of the Dupin indicatrix is
2
1
k x + k y = ±1.
2
2
1
2
There are several cases, depending on the sign of k k = K, i.e., depending on the sign of LN – M .
2
1 2
(1) If LN – M > 0, then k and k have the same sign. This is the case of an elliptic point.
2
1
2
If k  k , and k > 0 and k > 0, we get the ellipse of equation
2
1
2
1
x 2  y 2  1,
1 1
k 1 k 2
LOVELY PROFESSIONAL UNIVERSITY 375