INTRODUCTION TO MANIFOLDS — III
Algebra of vector fields. Lie derivative
(
s
)
.
1.
Notations.
The space of all
C
∞
smooth vector fields on a manifold
M
is
denoted by
X
(
M
).
If
v
∈
X
(
M
) is a vector field, then
v
(
x
)
∈
T
x
M
’
R
n
is its
value at a point
x
∈
M
.
The
flow
of a vector field
v
is denoted by
v
t
:
∀
t
∈
R
v
t
:
M
→
M
is a smooth map (automorphism) of
M
taking a point
x
∈
M
into the point
v
t
(
x
)
∈
M
which is the
t
endpoint of an integral trajectory for the field
v
, starting
at the point
x
.
♣
Problem 1.
Prove that the flow maps for a field
v
on a compact manifold
M
form a oneparameter group:
∀
t, s
∈
R
v
t
+
s
=
v
t
◦
v
s
=
v
s
◦
v
t
,
and all
v
t
are diffeomorphisms of
M
.
♣
Problem 2.
What means the formula
d
ds
fl
fl
fl
fl
s
=0
v
s
=
v
and is it true?
2.
Star conventions.
The space of all
C
∞
smooth functions is denoted by
C
∞
(
M
). If
F
:
M
→
M
is a smooth map (not necessary a diffeomorphism), then
there appears a contravariant map
F
*
:
C
∞
(
M
)
→
C
∞
(
M
)
,
F
*
:
f
7→
F
*
f,
F
*
(
x
) =
f
(
F
(
x
))
.
If
F
:
M
→
N
is a smooth map between two different manifolds, then
F
*
:
C
∞
(
N
)
→
C
∞
(
M
)
.
Note that the direction of the arrows is reversed!
♣
Problem 3.
Prove that
C
∞
(
M
) is a commutative associative algebra over
R
with respect to pointwise addition, subtraction and multiplication of functions.
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 Spring '08
 PROTSAK
 Algebra, Geometry, Derivative, Manifold, Vector field, Tx M Rn

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