Page 141 - Intro to Tensor Calculus

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This equation has two roots λ 1 and λ 2 which satisfy

Eg − Ge Ef − Fe
λ 1 + λ 2 = − and λ 1 λ 2 = , (1.5.38)
Fg − Gf Fg − Gf
where Fg − Gf 6=0. The curvatures κ (1) ,κ (2) corresponding to the roots λ 1 and λ 2 are called the principal
curvatures at the point P. Several quantities of interest that are related to κ (1) and κ (2) are: (1) the principal
radii of curvature R i =1/κ i,i =1, 2; (2) H = 1 (κ (1) + κ (2) ) called the mean curvature and K = κ (1) κ (2)
2
called the total curvature or Gaussian curvature of the surface. Observe that the roots λ 1 and λ 2 determine
two directions on the surface

r ∂~ ∂~ r ∂~ ∂~ r
r
r
r
d~ 1 d~ 2
= + λ 1 and = + λ 2 .
du ∂u ∂v du ∂u ∂v
If these directions are orthogonal we will have
r
r
r
r
d~ 1 d~ 2 ∂~ ∂~ ∂~ r ∂~ r
· =( + λ 1 )( + λ 2 )= 0.
du du ∂u ∂v ∂u ∂v
This requires that
Gλ 1 λ 2 + F(λ 1 + λ 2 )+ E =0. (1.5.39)
It is left as an exercise to verify that this is indeed the case and so the directions determined by the principal
curvatures must be orthogonal. In the case where Fg − Gf =0 we have that F =0 and f = 0 because the
coordinate curves are orthogonal and G must be positive. In this special case there are still two directions
determined by the differential equations (1.5.37) with dv =0, du arbitrary, and du =0, dv arbitrary. From
the differential equations (1.5.37) we find these directions correspond to

e g
κ (1) = and κ (2) = .
E G
α β
α
We let λ = du α denote a unit vector on the surface satisfying a αβ λ λ =1. Then the equation (1.5.34)
ds
α β
α β
can be written as κ (n) = b αβ λ λ or we can write (b αβ − κ (n) a αβ )λ λ =0. The maximum and minimum
α
normal curvature occurs in those directions λ where
α
(b αβ − κ (n) a αβ )λ =0
and so κ (n) must be a root of the determinant equation |b αβ − κ (n) a αβ | =0 or

1 b 1 b
γ
b − κ (n)
|a αγ b αβ − κ (n) δ | = 1 2 = κ 2 − b αβ a αβ κ (n) + =0. (1.5.40)
2
β b 2 1 b − κ (n) (n) a
2
This is a quadratic equation in κ (n) of the form κ 2 − (κ (1) + κ (2) )κ (n) + κ (1) κ (2) =0. In other words the
(n)
γ
principal curvatures κ (1) and κ (2) are the eigenvalues of the matrix with elements b = a αγ b αβ . Observe that
β
from the determinant equation in κ (n) we can directly find the total curvature or Gaussian curvature which
α
is an invariant given by K = κ (1) κ (2) = |b | = |a αγ b γβ | = b/a. The mean curvature is also an invariant
β
obtained from H = 1 (κ (1) + κ (2) )= 1 αβ b αβ , where a = a 11 a 22 − a 12 a 21 and b = b 11 b 22 − b 12 b 21 are the
a
2 2
determinants formed from the surface metric tensor and curvature tensor components.