§ 1. Classical Differential Calculus
163
and so
Γ
k
ij = Γ
k
ji .
(3.15)
On the other hand,
Δ
p
ij = −Γ
p
ij
for the following reason. Set
u(h) = B(h, u) for h ∈ E and u ∈ E
∗ .
This gives a bilinear function of h and u, so that
D i {B [h(ξ), u(ξ)]} = B [D i h(ξ), u(ξ)] + B [h(ξ), D i u(ξ)]
for any functions h(ξ) and u(ξ). Since B(a j , a
p ) = δ
p
j by definition of the dual
basis,
0 = B (D i a j , a
p ) + B (a j , D i a
p ) = Γ
q
ij B (a q , a
p ) + Δ
p
ik B
a j , a
k
= Γ
q
ij δ
p
q + Δ
p
ik δ
k
j = Γ
p
ij + Δ
p
ij
as announced. Hence the final formula
∇ i T
r
pq = D i T
r
pq − Γ
j
ip T
r
jq − Γ
j
iq T
r
pj + Γ
r
ij T
j
pq ,
(3.16)
where D i = d/dξ
i . These are the famous Christoffel formulas (1869), which
generalize in an obvious way to tensor fields of arbitrary type.
Exercise. Are the Γ
k
ij components of a tensor field ?
Index calculators did not leave it at that for their aim was differential
geometry on “ non-Euclidean curved ” spaces, and in particular on vector
spaces where distances are computed by non-Euclidean formulas. Using their
terminology, and as it has already been alluded to in n
◦ 1, (ii), suppose that
at every point of E there is simple formula for calculating the length ds of
the vector connecting some x ∈ E to an infinitesimally near point x + dx .
Since in an infinitesimal neighbourhood of x we want the geometry of the
space to be, as a first approximation, like that of an Euclidean space, we take
the square ds
2 to be a quadratic form
ds
2 = g ij (ξ)dξ
i dξ
j
(3.17)
in the coordinates dξ
i of dx, with a function g ij = g ji depending on the point
considered; the right hand side must obviously always be > 0 for dx = 0, in
other words
g ij (ξ)h
i h
j > 0
(3.18)