Polynomial Regression on Riemannian Manifolds

For any smooth family of curves γ s (t), we have
[ d
ds γ s(t), d dt γ s(t)]
= [W, ˙γ s ] = 0
so
We also need the Leibniz rule
∇ W ˙γ = ∇ ˙γ W.
d
ds 〈X, Y 〉| s=0 = 〈∇ W X, Y 〉 + 〈X, ∇ W Y 〉,
And the Riemann curvature tensor
R(X, Y )Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [X,Y ] Z
∇ W ∇ ˙γ Z = ∇ ˙γ ∇ W Z + R(W, ˙γ)Z
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 12