Polynomial Regression on Riemannian Manifolds

Lagrange multiplier (adjoint) vector fields λ i along γ:
E ∗ (γ, {v i }, {λ i }) =
N∑
g i (γ(t i )) +
i=1
∑k−1
+
i=1
∫ T
0
∫ T
0
〈λ 0 , ˙γ − v 1 〉dt
〈λ i , ∇ ˙γ v i − v i+1 〉dt +
Euler-Lagrange for {λ i } gives forward system.
Vector field integration by parts:
∫ T
0
〈λ i , ∇ ˙γ v i 〉dt = [〈λ i , v i 〉] T 0 −
∫ T
0
∫ T
0
〈∇ ˙γ λ i , v i 〉dt
〈λ k , ∇ ˙γ v k 〉dt
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 9