26
GEORGE A. HAGEDORN
/
5
E bJxi
Ni(X)
=
0
E
hixi
c2X2 + ^3X3 e
4
X
4
+ 1/5X5
-e
4
X
4
+ i/
5
X
5
c
2
X
2
- id5X3
\
\
c
2
X
2
- Jd
3
X
3
- e
4
X
4
- 1/5X5
e
4
X
4
- 1/5X5 c
2
X
2
+ 1^3X3
-E6^'
- E bix
(2.7)
/
To prove these claims we first note that if we replace ipj(X) by ipj(X), where
^P0 = /C^i(X),
MX) = z3ij3(x) +
Z^MX),
MX) = /c,^3W,
with H 2 + |2:2|
with |^3|2 + k
4
| 2 = 1, and
then N\(X) is transformed into
/ b-X
Ni(X) =
0
0 b-X
c-X-id-X -l-X-i] -X
\e-X-i~f-X c-X + id-X
c-X + id-X e-X + i~f -X\
-e - X H- i / X c-X-id-X
-b-X 0
0
-b-X J
We show below that by making an appropriate choice of the ZJ, we can force c, d, e, and
/ to be mutually orthogonal (and all non-zero in generic situations). Once this is done, we
rotate the X2, X3, X4, and X5 coordinate axes, so that c, 5, e, and / point along the X2,
X3, X4, and X5, respectively. This proves the claims.
Arbitrarily choosing the z^'s is equivalent to arbitrarily choosing two matrices U\ G
677(2) and U2 677(2), so that
(MX)\
=
(MX)\
\fc(x)J
l
W2P0;
and
(h{x)\
\MX))
u
(^x)\
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