Differential Geometry 1
Homework 02
1. Let
M
be a smooth manifold. By a
smooth curve through p
in
M
we mean
a smooth map
γ
:
I
→
M
from some open interval (0
∈
)
I
⊂
R
such that
γ
(0) =
p
.
(a) Show that a tangent vector
ξ
γ
∈
T
p
M
is defined by the rule
f
∈
C
∞
(
M
)
⇒
ξ
γ
(
f
) = (
f
◦
γ
) (0)
.
(b) Show that each tangent vector to
M
at
p
has the form
ξ
γ
for a suitable
choice of
γ
.
(c) Show that if
f
∈
C
∞
(
M
) has a local minimum at
p
then
f
is annihilated
by each tangent vector to
M
at
p
.
2. Let
Y
be a (finitedimensional real) vector space of which
X
is a subspace.
Show explicitly that P(
X
) is a submanifold of the projective space P(
Y
).
Now assume that
X
has codimension one in
Y
and let
g
∈
C
∞
(P(
Y
)).
Extract as much information as possible from the hypothesis P(
X
) =
g

1
(0).
Solutions
(1a) It need only be verified that
ξ
γ
:
C
∞
(
M
)
→
R
is linear and satisfies
the Leibniz rule at
p
; this takes far less time to write than to type.
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 Spring '09
 Robinson
 Vector Space, Manifold, tangent vector, Ψb

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