1. Differentiable Manifolds 7
If (17,u) is a chart on M with x G t / , then u* : f *- fou induces an isomor-
phism of algebras C ^
(Rn,
R) = C£°(M,R), and thus also an isomorphism
Txu : TXM -
Tu(a0Rn,
given by (T^.X
X
)(/) = X
x
( / o u). So TXM is an
n-dimensional vector space.
We will use the following notation: u =
(ul,..., un),
so
u%
denotes the i-th
coordinate function on £/, and
&\x •= (TxuyH&lvw) =
(Txu)-l(u(x),ei).
So -J=^ \x € TXM is the derivation given by
d_\ t t \ -
djfou-1)
dxs
l
Prom (1.7) we have now
*?!*(/) = — s i — M
1
) ) -
T^.X, = ^(T^.X,)^)^!^) = £*(** °u)^|tt(!r)
n
n
Xx = (Txu)-\Txu.Xx =
Y^X^u1)^.
i=l
1.9. The tangent bundle. For a manifold M of dimension n we put
TM := \_\xeM TXM, the disjoint union of all tangent spaces. This is a family
of vector spaces parameterized by M, with projection -KM • TM — M given
by irM(TxM) = x.
For any chart (Ua,ua) of M consider the chart {TxJ^{Ua),Tua) on TM,
where Tua : -Kj^(Ua) — ua(Ua) x
Rn
is given by
Tu
a
.X = (^(TTMpO^T^pQT/a.X).
Then the chart changings look as follows:
Tup o
(Tua)-1
: Tua(7r^(Ua(3)) = ua(Ua0) x
Rn
- •
-v u ^ t / ^ ) x
Mn
= Tupi^iU^)),
({Tufio(Tua)-l)(v,Y))U)
= ((Tua)-\y,Y)){foup)
= (v,Y)(f o up o
u'1)
=
d(foUpoU~1)(y).Y
= df{up o
u~l(y)).d(up
o u~
1
)(y).r
= (n
/ 3
ot
i
-
1
(y),d(n
/ 3
o«-
1
)(y).y)(/).
So the chart changings are smooth. We choose the topology on TM in such
a way that all Tua become homeomorphisms. This is a Hausdorff topology,
since X, Y E TM may be separated in M if 7r(X) ^ nCY); and they may be
