INTRODUCTION TO MANIFOLDS — II
Tangent Bundles
1. Tangent vectors, tangent space.
Let
M
n
be a smooth
n
dimensional manifold, endowed with an atlas of charts
x
:
U
→
R
n
,
y
:
V
→
R
n
,
. . .
, where
M
=
U
∪
V
∪ · · ·
are domains of the corre
sponding charts.
♥
Definition.
Two smooth curves
ϕ
i
: (

ε, ε
)
→
M
,
i
= 1
,
2, passing through
the same point
p
∈
M
, are said to be
1equivalent
,
ϕ
1
∼
ϕ
2
, if in some chart
x
:
U
→
R
n
k
x
(
ϕ
1
(
t
))

x
(
ϕ
2
(
t
))
k
=
o
(
t
)
as
t
→
0
+
.
(1)
♣
Problem 1.
Prove that the condition (1) is actually independent of the choice
of the chart.
♥
Definition.
The
tangent space
to the manifold
M
at the point
p
is the quo
tient space
C
1
(
R
1
, M
)
/
∼
by the equivalence (1).
Notations
: the equivalence class of a curve
ϕ
will be denoted by [
ϕ
]
p
. Instead
of saying that a curve
ϕ
belongs to a certain equivalence class
v
= [
·
]
p
, we say that
the curve
ϕ
is tangent to the vector
v
.
♣
Problem 2.
Prove that the tangent space at each point is isomorphic to the
arithmetic space
R
n
.
Solution.
Fix any chart
x
around the point
p
and consider the maps
f
iso
,
iso
defined as
f
iso:
R
n
→
C
∞
(
R
, M
)
,
v
= (
v
1
, . . . , v
n
)
7→
ϕ
v
(
·
)
,
ϕ
v
(
t
) =
x

1
(
x
(
p
) +
tv
) (2)
iso(
v
) = [
f
iso(
v
)]
p
.
(3)
This map is injective (prove!).
To prove its surjectivity, for any smooth curve
ϕ
consider its
x
coordinate representation,
x
(
ϕ
(
t
)) =
x
(
p
) +
tv
+
· · ·
,
existing by virtue of differentiability of the latter. Then
f
iso(
v
)
∼
ϕ
.
Remark.
This is a good example of abstract nonsense! The idea is that you associate
with each curve its linear terms, the coordinate system being fixed. Then any curve
is uniquely defined by its linear terms up to the 1equivalence, since the definition
(1) was designed especially for this purpose!
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TANGENT BUNDLES
♥
Definition.
The string of real numbers (
v
1
, . . . , v
n
) is called the coordinate
representation of the tangent vector [
ϕ
]
p
in the coordinate system
x
.
♣
Problem 3.
If
M
n
is a hypersurface in
R
n
+1
, then the tangent space is well de
fined by geometric means. Prove that this “geometric” tangent space is isomorphic
to the one defined by the abstract definition above. A good exercise for practicing
in abstract nonsense!
Important note:
The coordinate system
x
occurs in the construction of iso
morphisms (2), (3) in the most essential way!
If another coordinate system is
chosen, then the isomorphisms will be completely different.
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 Spring '08
 PROTSAK
 Geometry, Vectors, Vector Space, Manifold, Vector field, Euclidean space Rn, tangent bundles, arithmetic space Rn

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