Dmitry Fuchs
DIFFERENTIAL GEOMETRY (240A)
1. Manifolds.
1.1 Definition of a manifold.
1.1.1. Charts.
Let
M
be a set. An
n
dimensional chart
on
M
is a pair (
U, ϕ
) where
U
is an open
subset of
R
n
and
ϕ
is a 1–1 map of
U
into
M
.
Two
n
dimensional charts on
M,
(
U, ϕ
) and (
V, ψ
), are called
compatible
, if
(1) the set
ϕ

1
(
ψ
(
V
))
⊂
U
is open (in
U
⇒
in
R
n
);
(2) the set
ψ

1
(
ϕ
(
U
))
⊂
V
is open (in
V
⇒
in
R
n
);
(3) the map
ϕ

1
(
ψ
(
V
))
u
7→
ψ

1
ϕ
(
u
)
→
ψ

1
(
ϕ
(
U
)) is smooth (=
C
∞
⇒
continuous);
(4) the map
ψ

1
(
ϕ
(
U
))
v
7→
ϕ

1
ψ
(
v
)
→
ϕ

1
(
ψ
(
V
)) is smooth.
In particular, (
U, ϕ
) and (
V, ψ
) are compatible if
ϕ
(
U
)
∩
ψ
(
V
) =
∅
.
1.1.2. Atlases.
A set
{
(
U
α
, ϕ
α
)

α
∈
A
}
of
n
dimensional charts on
M
is called an (
n
dimensional)
atlas
if
(1)
[
α
∈
A
ϕ
α
(
U
α
) =
M
;
(2) For any
α, β
∈
A
the charts (
U
α
, ϕ
α
)
,
(
U
β
, ϕ
β
) are compatible.
Two
n
dimensional atlases on
M
,
A
and
B
are called equivalent, if their union
A ∪ B
is also an atlas (in other words, if any chart of
A
is compatible with any chart of
B
.
1.1.3. Topology.
Let
M
be a set with an
n
dimensional atlas
A
. A subset
B
of
M
is called open (with
respect to
A
), if for any chart (
U, ϕ
)
∈ A
the set
ϕ

1
(
B
) is open (in
U
⇒
in
R
n
).
(In
particular, the sets
ϕ
(
U
) are open.)
Proposition.
If the atlases
A
and
B
are equivalent, then a set
B
⊂
M
is open with
respect to
A
if and only if it is open with respect to
B
.
Proof
. For any
B
⊂
M
,
B
=
[
(
V,ψ
)
∈
B
(
B
∩
ψ
(
V
)) =
[
(
V,ψ
)
∈
B
ψ
(
ψ

1
(
B
))
.
If
B
is open with respect to
B
, then all the sets
ψ

1
(
B
) are open. Let (
U, ϕ
) be a chart of
A
. Then
ϕ

1
(
B
) =
ϕ

1
[
(
V,ψ
)
∈
B
ψ
(
ψ

1
(
B
)) =
[
(
V,ψ
)
∈
B
ϕ

1
ψ
(
ψ

1
(
B
)) =
[
(
V,ψ
)
∈
B
(
ψ

1
ϕ
)

1
(
ψ

1
(
B
))
.
1