Math 497C
Nov 11, 2004
1
Curves and Surfaces
Fall 2004, PSU
Lecture Notes 14
2.11 The Induced Lie Bracket on Surfaces; The Self
Adjointness of the Shape Operator Revisited
If
V
,
W
are tangent vectorFelds on
M
, then we deFne
[
V,W
]
M
:=
∇
V
W
−∇
W
V,
which is again a tangent vector Feld on
M
. Note that since, as we had veriFed
in an earlier exercise,
S
is selfadjoint, the Gauss’s formula yields that
[
V,W
]=
∇
V
W
−
∇
W
V
=
∇
W
V
−∇
V
W
+
³

V,S
(
W
)
®
−

W, S
(
V
)
®
´
n
=[
V,W
]
M
.
In particular if
V
and
W
are tangent vectorFelds on
M
, then [
V,W
] is also
a tangent vectorFeld.
Let us also recall here, for the sake of completeness, the proof of the self
adjointness of
S
. To this end it suﬃces to show that if
E
i
,
i
=1
,2
,i
sa
basis for
T
p
M
, then
h
E
i
,S
p
(
E
j
)
i
=
h
S
p
(
E
i
)
,E
j
i
. In particular we may let
E
i
=
X
i
(0
,
0), where
X
:
U
→
M
is a regular patch of
M
centered at
p
.Now
note that

X
i
,S
p
(
X
j
)
®
=
−

X
i
,dn
p
(
X
j
)
®
=
−

X
i
,
(
n
◦
X
)
j
®
=

X
ij
,
(
n
◦
X
)
®
.
Since the right hand side of the above expression is symmetric with respect
to
i