20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates 423
Example 20.2 (Geodesic as a submanifold in {E
2
A 1 ,A 2
, G KL }, B = c = const: “parallel circle”).
X = +E 1
A 1 cos c cos T
1 − E 2 sin
2 c
+ E 2
A 1 cos c sin T
1 − E 2 sin
2 c
+ E 3
A 1 (1 − E
2 ) sin c
1 − E 2 sin
2 c
,
(20.38)
D 1 = −E 1 sin T + E 2 cos T =: ˙
X/ ˙
X ,
D 2 = −E 1 sin c cos T − E 2 sin c sin T + E 3 cos c ,
D 3 = +E 1 cos c cos T + E 2 cos c sin T + E 3 sin c ,
(20.39)
D =
⎡
⎢
⎢
⎣
− sin T
cos T
0
− sin c cos T − sin c sin T cos c
cos c cos T
cos c sin T sin c
⎤
⎥
⎥
⎦ E ,
D = RE , D
= R
E = R
R
T D = ΩD ∀ R ∈ SO(3) ,
R
=
⎡
⎢
⎢
⎣
−T
cos T
−T
sin T
0
+T
sin c sin T −T
sin c cos T 0
−T
cos c sin T +T sin c cos T 0
⎤
⎥
⎥
⎦ ,
Ω := R
R
T =
⎡
⎢
⎢
⎣
0
+T
sin c −T
cos c
−T
sin c
0
0
+T
cos c
0
0
⎤
⎥
⎥
⎦ ,
(20.40)
κ g = +T
sin c , κ n = −T
cos c , τ g = 0 ,
˙
X =
dS
dT
=
A 1 cos c
1 − E 2 sin
2 c
,
T
:=
dT
dS
=
1 − E 2 sin
2 c
A 1 cos c
⇒
κ g = +
1 − E 2 sin
2 c
A 1
tan c = const ,
κ n = −
1 − E 2 sin
2 c
A 1
= const ,
τ g = 0 .
(20.41)
End of Example.
Obviously, the parallel circle B = const is a geodesic circle. Note that for a sphere S
2
R great circles
are geodesics, but small circles are geodesic circles. Following the curvature measure representation in
Corollary 20.1, we can characterize geodesics and geodesic circles by differential equations.