Page 144 - Intro to Tensor Calculus

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where J is the Jacobian of the surface coordinate transformation. Here the curvature tensor for the surface
R αβγδ has only one independent component since R 1212 = R 2121 = −R 1221 = −R 2112 (See exercises 20,21).
From the transformation law
γ
α
β
∂u ∂u ∂u ∂u δ
¯
R ηλµ = R αβγδ η λ µ
∂¯u ∂¯u ∂¯u ∂¯u
2
¯
one can sum over the repeated indices and show that R 1212 = R 1212 J and consequently
¯
R 1212 R 1212
= = K
¯ a a
which shows that the Gaussian curvature is a scalar invariant in V 2 .
Geodesic Curvature

~
For C an arbitrary curve on a given surface the curvature vector K, associated with this curve, is
the vector sum of the normal curvature κ (n) bn and geodesic curvature κ (g) bu and lies in a plane which
is perpendicular to the tangent vector to the given curve on the surface. The geodesic curvature κ (g) is
obtained from the equation (1.5.25) and can be represented

!
~
~
dT dT ~ dT
~ ~ ~
κ (g) = bu · K = bu · =(bn × T) · = T × · bn.
ds ds ds
Substituting into this expression the vectors
d~ du dv
r
~
r
r
T = = ~ u + ~ v
ds ds ds
dT ~ 0 2 0 2
~
0
0
00
= K = ~ uu (u ) +2~r uv u v + ~r vv (v ) + ~r u u + ~r v v ,
00
r
ds
where 0 = d , and by utilizing the results from problem 10 of the exercises following this section, we find
ds
that the geodesic curvature can be represented as

2 0 3 2 1 0 2
0
κ (g) = (u ) + 2 − (u ) v +
11 12 11
(1.5.48)

2 1 0 2 1 0 3 p
2
0
0
00
0
00
− 2 u (v ) − (v ) +(u v − u v ) EG − F .
22 12 22
This equation indicates that the geodesic curvature is only a function of the surface metrices E, F, G and
0
00
0
the derivatives u ,v ,u ,v . When the geodesic curvature is zero the curve is called a geodesic curve. Such
00
curves are often times, but not always, the lines of shortest distance between two points on a surface. For
example, the great circle on a sphere which passes through two given points on the sphere is a geodesic curve.
If you erase that part of the circle which represents the shortest distance between two points on the circle
you are left with a geodesic curve connecting the two points, however, the path is not the shortest distance
between the two points.
For plane curves we let u = x and v = y so that the geodesic curvature reduces to
dφ
00
0
k g = u v − u v =
0
00
ds