Page 64 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 53

Moreover, we can have surface covariant and contravariant second-order tensors with
covariant derivatives given by formulas corresponding to eqns (1.126) and (1.129), but
with Greek rather than Roman indices. In particular, for the contravariant metric tensor
αβ
a , we can deduce that its surface covariant derivatives are
α
β
∂a αβ ∂(a · a )
αβ α δβ β αδ α δβ β αδ
a ,γ = + a + a = + a + a
δγ
δγ
δγ
δγ
∂u γ ∂u γ
∂a β ∂a α
α β α δβ β αδ
= a · + · a + a + a
δγ
δγ
∂u γ ∂u γ
β αδ α δβ α δβ β αδ
=− a − a + a + a = 0, (3.59)
δγ δγ δγ δγ
where we have made use of the surface equivalent of eqn (1.105). Similarly we can
show that
a αβ,γ = 0 (3.60)
for all α, β, γ ranging over the values 1, 2.
These results are equivalent to those for the metric tensor for general curvilinear co-
ordinates in Chapter 1. Note, however, that the supplementary argument given there
based on the existence of a background rectangular cartesian system in which all
the covariant derivatives are necessarily zero no longer applies, since a general two-
dimensional surface does not admit a cartesian system (unless it is planar).
The surface equation analogous to eqn (1.118) is
1 ∂ √
α
αβ = √ ( a), (3.61)
a ∂u β
and this may be used in the process of proving that the covariant derivatives of ε αβ
and ε αβ satisfy
ε αβ = ε αβ,γ = 0. (3.62)
,γ
Investigation of the commutativity of repeated covariant differentiation of an arbi-
trary surface covariant vector A α leads to the equation

A α,βγ − A α,γβ = R δ A δ , (3.63)
.αβγ
as in eqn (1.177), where the mixed surface Riemann-Christoffel tensor is given by
δ
∂ δ αγ ∂ αβ µ
R δ .αβγ = − + µ δ µβ − δ (3.64)
αγ
αβ µγ
∂u β ∂u γ
similarly to eqn (1.178). Moreover, we can define the covariant fourth-order surface
curvature tensor
µ
R αβγ δ = a αµ R . (3.65)
.βγ δ
Note that these fourth-order surface tensors do not in general vanish identically,
unlike their three-dimensional counterparts in a Euclidean space, as considered in
Chapter 1. They have the same skew-symmetric properties discussed there, however,
and it follows that the only possible non-vanishing components of R αβγ δ are
R 1212 = R 2121 =−R 1221 =−R 2112 .