Page 28 - Curvature and Homology

10 I. RIEMANNIAN MANIFOLDS
the corresponding space of tensors of type (Y, s)., If we fix a base in v,
a base of T' is determined. Let U be a coordinate neighborhood and u
the corresponding homeomorphism from U to En. The local coordinates
of a point P E U will be denoted by (ui(P)); they determine a base
{dui(P)} in T: and a dual base {ei(P)} in Tp. These bases give rise to a
well-defined base in T,Z(P). Consider the map

where yU(P, t), P E U, t E ?''; belongs to TI(P) and has the same com-
ponents fi...irjl:.Jl relative to the (natural) base of T'(P) as t has in c.
That y, is 1-1 is clear. Now, let V be a second coordinate neighborhood
such that U n V # (the empty set), and consider the map

is a 1-1 map of ?",Z onto itself. Let (v!(P)) denote the local coordinates of P
in V. They determine a base {dv"P)} in T,* and a dual base Cfi(P)}
in Tp. If we set
f = gcrv(p)t, (1.3.3)
it follows that
c~v(P,i) = ~v(P,t). (1.3.4)
Since
V~(P,F)
= &*--ir jl...ja eil...ir il...ia (p)
(1.3.5)
and
~v(P,t) jl...jH ftl,..irjl-.j~ p) (1 J.6)
= &.-ir
where {eil.. ,,jl..Ja(P)} and { fil.. ,"...jl(P)) are the induced bases in
w'),

These are the equations defining gUv(P). Hence gu,(P) is a linear
automorphism of TI. If we give to Tl the topology and differentiable