It turns out that the smooth structure that
T M
inherits is directly from
M
itself so that the projection
π
is a smooth map, and so that for any smooth
F : M → N , the following diagram commutes:
T M T N
M N
F
∗
π π
F
Moreover, this
F
∗
is a smooth map with respect to the smooth structures of
T M
and T N.
The thing that is not obvious from our deﬁnition is that with respect to the
smooth structure, the tangent bundle
T M
of an
n
-manifold
M
is not necessarily
diﬀeomorphic to
M × R
n
. It is true that the only two tangent bundles we have
deﬁned—for
R
n
and
S
1
—have been diﬀeomorphic to this “trivial” bundle, but
in general it does not hold. If it did, one might guess that the topic of tangent
bundles would be really bland and boring!
The idea behind the tangent bundle is that of the more general “vector
bundles”. A vector bundle of rank n is a triple (E, B, π) where:
(i) E and B are topological spaces.
(ii) π : E → B is a continuous surjective map.
(iii)
For each
x ∈ B
,
π
−1
(
x
) has the structure of an
n
-dimensional vector space.
(iv)
For each
x ∈ B
there exists an open neighbourhood
U
of
x
such that
π
−1
(U) is homeomorphic to U × R
n
.
The ﬁnal condition states that every point
x ∈ B
has a neighbourhood which is
“trivial” with respect to the vector bundle—that when you look at
π
−1
(
U
) for a
certain
U
, it looks like a “bundle” of copies of
R
n
, one for each
y ∈ U
. If a global
trivializing neighbourhood U = B exists, then the bundle is called trivial.
In our case, we are considering the smooth vector bundles (
T M, M, π : T M →
M
) where we additionally require smoothness instead of just continuity. An
example of where
T M 6
∼
=
M × R
n
is the case for
M
=
S
2
(i.e. non-trivial). This
is not obvious, but relates to the (non)existence of a non-vanishing vector ﬁeld
and the “hairy ball” theorem
6
.
Intuitively, a vector ﬁeld is a map which takes a point in
M
and spits out a
tangent vector. Hence, a “ﬁeld of vectors” on
M
. Formally, a
vector ﬁeld
is
deﬁned to be a smooth map
v : M → T M
such that
π ◦ v
=
id
M
(that is, it is a
smooth “section” of π).
Example 1.4. Consider M = R
2
, deﬁning v : R
2
→ T R
2
by
v(x, y) := v
(x,y)
= (−y, x),
6
One cannot comb a hairy ball without making a cowlick!
9