Page 37 - Basic Structured Grid Generation

26 Basic Structured Grid Generation

1.9 The Riemann-Christoffel tensor

The covariant third-order tensor u i,jk formed from the covariant components u i of
a vector by using eqn (1.126) to obtain the covariant derivatives of the covariant
second-order tensor u i,j given by eqn (1.122) is found to be

2 ∂ l
∂ u i l ∂u l l ∂u i l ∂u l ij l m l m
u i,jk = − ij − jk − ik − u l + u m + u m .
jk il
ik lj
k
∂x ∂x j ∂x k ∂x l ∂x j ∂x k
(1.176)
We can investigate the commutativity of successive covariant differentiations by
subtracting from this expression a similar one with j and k interchanged. This gives
l l
∂ ∂ ij
m l
m l
u
u i,jk − u i,kj = ik − + − mk u l = R l .ijk l , (1.177)
ij
ik mj
∂x j ∂x k
where
∂ l ∂ l ij
m l
m l
R l = ik − + − . (1.178)
.ijk j k ik mj ij mk
∂x ∂x
The left-hand side of eqn (1.177) represents a covariant third-order tensor, and it
follows that, for the right-hand side also to represent a tensor, R l must beamixed
.ijk
fourth-order tensor. It is called the Riemann-Christoffel tensor. In fact, since our back-
ground space is Euclidean and the Christoffel symbols all vanish when we take a
rectangular cartesian set of co-ordinates, all the components of R l will also be zero
.ijk
in cartesian co-ordinates, and, moreover, will always transform to zero under tensor
transformation rules for any other choice of co-ordinates. We may also prove that
l
R .ijk = 0 (1.179)
3
in E for all i, j, k, l, by substituting directly for l from eqn (1.102) (which also
ik
assumes the existence of a background cartesian system) into eqn (1.178).
It will not be necessary here to consider non-Euclidean three-dimensional spaces,
in which non-vanishing Riemann-Christoffel tensors may exist, but in Chapter 3 we
shall need to consider a two-dimensional version of eqn (1.178) on an arbitrary curved
surface within a three-dimensional Euclidean space. So we now set out some general
results for three-dimensional non-Euclidean Riemann-Christoffel tensors (as defined by
eqn (1.178)) to serve as a basis for comparison with the Riemann-Christoffel tensor
defined on a curved two-dimensional surface, to appear later. First we note that the
covariant associated tensor R ijkl , called the curvature tensor,given by
m
R ijkl = g im R ..jkl (1.180)
ij
may be defined. Using eqns (1.100), (1.108), the symmetry of g ij and g ,and the
j
relation g im g jm = δ , it may be shown that
i

2
2
2
2
1 ∂ g il ∂ g jk ∂ g ik ∂ g jl
R ijkl = + − −
j
j
i
i
2 ∂x ∂x k ∂x ∂x l ∂x ∂x l ∂x ∂x k
mn
+g ([jk, m][il, n]− [jl, m][ik, n]). (1.181)