Polynomial Regression on Riemannian Manifolds

Now do the same with another term T i
d
ds T i(γ s )| s=0 = d ds
=
∫ T
0
= 0 +
= 0 +
∫ T
0
〈λ i , ∇ ˙γ v i 〉dt
〈∇ W λ i , ∇ ˙γ v i 〉 + 〈λ i , ∇ W ∇ ˙γ v i 〉dt
∫ T
0
∫ T
0
〈λ i , ∇ ˙γ ∇ W v i + R(W, ˙γ)v i 〉dt
〈R(λ i , v i ) ˙γ, W 〉dt
where we used Bianchi identities to rearrange the curvature
term. So
δ γ(t) T i = R(λ i , v i ) ˙γ
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 14