20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates 421
Box 20.2 (Surface geometry of E
2
A 1 ,A 2 ).
Matrices
`
Frobenius matrix F (elements a, b, c, d), Gauss matrix G = J
T J, (elements e, f , g),
Hesse matrix H =
˘˙
X , KL , G 3
¸¯
(elements l, m, n), curvature matrix K,
Jacobi matrix J =
˘
∂X
J /∂U
K
¯´
:
F = {F
2
KL } =
(1−E
2 sin
2 U cos
2 V )
1/2
A 1 (1−sin 2 U cos 2 V ) 1/2
2
4
− sin U sin V
cos V
cos U (1−E
2 sin
2 U cos
2 V )
(1−E 2 ) cos V
− cos U
− sin U sin V (1−E
2 sin
2 U cos
2 V )
(1−E 2 )
3
5 ,
(20.21)
G = {G
2
KL } =
=
»
G 11 G 12
G 21 G 22
–
,
8
> > > > <
> > > > :
G 11 =
A
2
1 cos
2 V (1−2E
2 (1−sin
2 U sin
2 V )+E
4 (1−sin
2 U sin
2 V (1+sin
2 U cos
2 V )))
(1−E 2 sin 2 U cos 2 V ) 3
,
G 12 = G 21 =
A
2
1 E
2 cos U sin U cos V sin V (2−E
2 (1+cos
2 V sin
2 U ))
(1−E 2 sin 2 U cos 2 V ) 3
,
G 22 =
A
2
1 (1−2E
2 sin
2 U +E
4 sin
2 U (1−cos
2 V cos
2 U ))
(1−E 2 sin 2 cos 2 V ) 3
,
(20.22)
H = {H
2
KL } =
2
6
4
−A 1 cos
2 V (1−E
2 (1−sin
2 U sin
2 V ))
(1−E 2 sin 2 U cos 2 V ) 3/2
−A 1 E
2 sin U cos U sin V cos V
(1−E 2 sin 2 U cos 2 V ) 3/2
symmetric
−A 1 (1−E
2 sin
2 U )
(1−E 2 sin 2 U cos 2 V ) 3/2
3
7
5 ,
(20.23)
K = {K
2
KL } =
=
2
6
4
(1−E
2 sin
2 U )(1−E
2 sin
2 U cos
2 V )
1/2
A 1 (1−E 2 )
− cos U cos V sin U sin V E
2 (1−E
2 sin
2 U cos
2 V )
1/2
A 1 (1−E 2 )
− cos U sin U sin V E
2 (1−E
2 sin
2 U cos
2 V )
1/2
A 1 (1−E 2 ) cos V
(1−E
2 (1−sin
2 U sin
2 V ))(1−E
2 sin
2 U cos
2 V )
1/2
A 1 (1−E 2 )
3
7
5 ,
(20.24)
h = −
tr[K]
2
= −
√
1−E 2 sin 2 U cos 2 V (2−E
2 (1+sin
2 U cos
2 V ))
2A 1 (1−E 2 )
, k = det[K] =
(1−E
2 sin
2 U cos
2 V )
2
A 2
1
(1−E 2 )
, (20.25)
J = {J
2
KL } =
2
6
6
6
6
4
A 1 cos V sin U (1−E
2 cos
2 V )
(1−E 2 sin 2 U cos 2 V ) 3/2
−A 1 cos U sin V
(1−E 2 sin 2 U cos 2 V ) 3/2
A 1 (1−E
2 ) cos V cos U
(1−E 2 sin 2 U cos 2 V ) 3/2
−A 1 (1−E
2 ) sin U sin V
(1−E 2 sin 2 U cos 2 V ) 3/2
A 1 E
2 sin U cos U sin V cos
2 V
(1−E 2 sin 2 U cos 2 V ) 3/2
A 1 cos V (1−E
2 sin
2 U )
(1−E 2 sin 2 U cos 2 V ) 3/2
3
7
7
7
7
5
.
(20.26)
Eigenvalues:
1st eigenvalue of K: κ 1 =
p
1 − E 2 sin
2 U cos 2 V /A 1 ;
2nd eigenvalue of K: κ 2 = (1 − E
2 sin
2 U cos
2 V )
3/2 /A 1 (1 − E
2 ).
Christoffel symbols
˘ M
KL
¯
:
{
1
11 }(U, V ) =
E
2 sin U cos U cos
2 V (3−E
2 (3−sin
2 U sin
2 V ))
(1−E 2 sin 2 U cos 2 V )(1−E 2 )
,
{
1
12 }(U, V ) = −
sin V (1−E
2 −E
4 sin
2 U cos
2 U cos
2 V )
(1−E 2 sin 2 U cos 2 V )(1−E 2 ) cos V
, {
1
22 }(U, V ) =
E
2 sin U cos U (1−E
2 sin
2 U )
(1−E 2 sin 2 U cos 2 V )(1−E 2 )
,
{
2
11 }(U, V ) =
sin V cos V (1−E
2 (1+2 sin
2 U cos
2 V )+E
4 sin
2 U cos
2 V (2−sin
2 U sin
2 V ))
(1−E 2 sin 2 U cos 2 V )(1−E 2 )
{
2
12 }(U, V ) =
E
2 sin U cos U cos
2 V (1−E
2 (1+sin
2 U sin
2 V ))
(1−E 2 sin 2 U cos 2 V )(1−E 2 )
, {
2
22 }(U, V ) =
−E
2 sin
2 U sin V cos V (3−E
2 (2+sin
2 U ))
(1−E 2 sin 2 U cos 2 V )(1−E 2 )
.
(20.27)