Discrete Curve Flows in Two-Dimensional Cayley–Klein Geometries
161
K n = ι n (x n+1 ) =
|Δz n |
|1 + 1 z n z n+1 |
,
I n + i 2 J n = ι n (z n−1 ) = −
Δz n Δz n−1 (1 + 1 z n z n+1 )(1 + 1 z n z n−1 )
|Δz n ||1 + 1 z n z n+1 ||1 + 1 z n z n−1 | 2 ,
(3)
where I n , J n are the real and imaginary parts of ι n (z n−1 ), respectively. The
invariantization map ι n extends to one-forms [1]. For example, the invariantization
of dz k = dx k + i 2 dy k is the invariant one-form
k
n = ω
k
n + i 2 σ
k
n = ι n (dz k ) =
Δz n (1 + 1 z n z n )(1 + 1 z n z n+1 )(1 + 1 z n z k ) 2 dz k
|Δz n ||1 + 1 z n z n+1 ||1 + 1 z n z k | 4
,
where ω k
n , σ k
n are the real and imaginary parts of ι n (dz k ), respectively.
4 Recurrence Relations
We now compute the recurrence relations for the exterior derivative and the shift
map. These formulas encapsulate the fact that the invariantization map does not,
in general, commute with the exterior derivative and the shift map. The general
recurrence formulas can be found in [1]. Below, we specialize these formulas to our
problem.
4.1 Exterior Derivative
Let μ 1 , μ 2 , and μ 12 be a basis of Maurer–Cartan forms dual to the infinitesimal
generators (2). Then the recurrence relations for the exterior derivative are
d[ι n (x k )] = ω
k
n + [1 + 1 (ι n (x k )
2
− 2 ι n (y k )
2
]ν 1
+ 2 1 2 ι n (x k )ι n (y k )ν 2 − 2 ι n (y k )ν 12 ,
d[ι n (y k )] = σ
k
n +2 1 ι n (x k )ι n (y k )ν 1 +[1− 1 (ι n (x k )
2
− 2 ι n (y k )
2
]ν 2 +ι n (x k )ν 12 ,
(4)
where ν 1 = ρ ∗ μ 1 , ν 2 = ρ ∗ μ 2 , ν 12 = ρ ∗ μ 12 denote the pull-back of the Maurer–
Cartan forms via the moving frame. The recurrence relations for the phantom
invariants ι n (x n ) = ι n (y n ) = ι n (y n+1 ) = 0 yield the normalized Maurer–Cartan
forms
ν 1 = −ω
n
n ,
ν 2 = −σ
n
n ,
ν 12 =
σ n
n − σ n+1
n
K n
− 1 K n σ
n
n .
161
K n = ι n (x n+1 ) =
|Δz n |
|1 + 1 z n z n+1 |
,
I n + i 2 J n = ι n (z n−1 ) = −
Δz n Δz n−1 (1 + 1 z n z n+1 )(1 + 1 z n z n−1 )
|Δz n ||1 + 1 z n z n+1 ||1 + 1 z n z n−1 | 2 ,
(3)
where I n , J n are the real and imaginary parts of ι n (z n−1 ), respectively. The
invariantization map ι n extends to one-forms [1]. For example, the invariantization
of dz k = dx k + i 2 dy k is the invariant one-form
k
n = ω
k
n + i 2 σ
k
n = ι n (dz k ) =
Δz n (1 + 1 z n z n )(1 + 1 z n z n+1 )(1 + 1 z n z k ) 2 dz k
|Δz n ||1 + 1 z n z n+1 ||1 + 1 z n z k | 4
,
where ω k
n , σ k
n are the real and imaginary parts of ι n (dz k ), respectively.
4 Recurrence Relations
We now compute the recurrence relations for the exterior derivative and the shift
map. These formulas encapsulate the fact that the invariantization map does not,
in general, commute with the exterior derivative and the shift map. The general
recurrence formulas can be found in [1]. Below, we specialize these formulas to our
problem.
4.1 Exterior Derivative
Let μ 1 , μ 2 , and μ 12 be a basis of Maurer–Cartan forms dual to the infinitesimal
generators (2). Then the recurrence relations for the exterior derivative are
d[ι n (x k )] = ω
k
n + [1 + 1 (ι n (x k )
2
− 2 ι n (y k )
2
]ν 1
+ 2 1 2 ι n (x k )ι n (y k )ν 2 − 2 ι n (y k )ν 12 ,
d[ι n (y k )] = σ
k
n +2 1 ι n (x k )ι n (y k )ν 1 +[1− 1 (ι n (x k )
2
− 2 ι n (y k )
2
]ν 2 +ι n (x k )ν 12 ,
(4)
where ν 1 = ρ ∗ μ 1 , ν 2 = ρ ∗ μ 2 , ν 12 = ρ ∗ μ 12 denote the pull-back of the Maurer–
Cartan forms via the moving frame. The recurrence relations for the phantom
invariants ι n (x n ) = ι n (y n ) = ι n (y n+1 ) = 0 yield the normalized Maurer–Cartan
forms
ν 1 = −ω
n
n ,
ν 2 = −σ
n
n ,
ν 12 =
σ n
n − σ n+1
n
K n
− 1 K n σ
n
n .