544 E Geodesics
Proof (Corollary 20.1).
First statement:
({
M
KL }U
K U
L + U
M )G M = κ g D 2
=⇒
κ
2
g = κ g D 2 |κ g D 2 =
= ({
M 1
K 1 L 1
}U
K 1 U
L 1 + U
M 1 )G M 1 M 2 ({
M 2
K 2 L 2
}U
K 2 U
L 2 + U
M 2 )
=⇒
(20.28) .
(E.6)
Second statement:
=⇒
κ n = H KL U
K U
L
=⇒
(20.29) .
(E.7)
Third statement:
D
1 = κ
g D 2 + κ g D
2 + κ
n D 3 + κ n D
3 ,
D
2 = −κ g D 1 + τ g D 3
=⇒
D
1 = −(κ
2
g + κ
2
n )D 1 + (κ
g − κ n τ g )D 2 + (κ g τ g + κ
n )D 3 ,
(E.8)
D 1 = G K U
K ,
D
1 = G K,L U
K U
L + G K U
K ,
D
1 = G K,LM U
K U
L U
M + G K,L (U
K U
L + U
K U
L ) + G K,L U
K U
L + G K U
K ,
(E.9)
G K,LM =
= {
N
KL }, M G N + {
N
KL }G N,M + H KL,M G 3 + H KL G 3,M ,
G K,LM =
= −H KL H MM 2 G
M 2 K 2 G K 2 +
+{
N 1
KL }{
M 2
N 1 M }G M 2 + {
N 1
KL }H N 1 ,M G 3 + {
N
KL }, M G N + H KL,M G 3 ,
(E.10)
D
1 =
= [U
M + (2U
K U
L + U
K U
L ){
M
KL } + U
K U
L U
M 2 ({
M
KL }, M 2 +{
M 1
KL }{
M
M 1 M 2
}−
−H KL H M 1 M 2 G
M 1 M )]G M + [(2U
K U
L + U
K U
L )H KL +
+U
K U
L U
M 2 ({
M 1
KL }H M 1 M 2 + H KL,M 2 )]G 3 .
(E.11)