Graphics Reference
In-Depth Information
Here are two alternate descriptions of the tangent bundle of a manifold M n . We
shall only define the new total spaces E in each case because the rest of the defini-
tions are obvious.
First alternate definition of t M :
Let
Ê
Á
ˆ
˜
U
n
E
=
U
¥
R
/~
(8.28)
coordinate neighborhoods
(
U
,
j
)
where the equivalence relation ~ is defined as follows: If ( U ,j)} and ( V ,y) are coordi-
nate neighborhoods for M n , then
) δ n
(
qa
,
U R
is identified with
(
(
)
) δ
-
1
(
()
)( )
n
q
,D
yj
o
j
q
a
V
R
(8.29)
for all q Œ U « V . Recall our comments about fiber bundles earlier. We can now see
how the group GL(n, R ) enters the picture, namely, through the linear isomorphism
D(yj -1 )(j( q )).
o
Second alternate definition of t M : This description is only applicable when M n is
a submanifold of some R k : In this case, using our definitions related to the manifolds
in Section 8.4, define
{
}
(
) δ
n
k
n
E , vMRv
=
is a
tangent
vector to
M p
at
.
(8.30)
In the terminology of Section 8.4, it could happen that two different points p and q
of M n could call the same v ΠR k as their tangent vector, but we differentiated between
the uses of v by the phrase “at p ” or “at q .” That was an adequate way to deal with
the distinction there, but in the context of abstract manifolds it is important that
tangent vectors at one point of a manifold are different from those at another point
and to have that fact incorporated into the definition. The definition of E in (8.30) is
the most convenient way to accomplish that. The pair ( p , v ) (as an element of T p ( M n ))
now formalizes the entire phase “the tangent vector v of M n at p .” As an added bonus
there is no need to worry about what topology to give E because E inherits a natural
topology as a subspace of M n ¥ R k . Note that if M n = R n , then the tangent space T p ( M n )
at a point p is just p ¥ R n , which agrees with the definition in Section 4.9.
Showing that these two new definitions of t M are equivalent to the original one is
left to the reader (Exercise 8.10.1). In the future we shall feel free to choose whichever
definition is most convenient.
8.10.2. Example. The tangent bundle of S 1 is trivial. We can see that from Figure
8.31. The oriented tangent lines to the circle in R 2 (Figure 8.31(a)) can be rotated into
a vertical direction in R 3
(Figure 8.31(b)), so that the total space is nothing but S 1
¥
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