Polynomial Regression on Riemannian Manifolds

Curvature on preshape sphere S 2N−1 , R(X, Y )Z is:
R(X, Y )Z = 〈X, Z〉Y − 〈Y, Z〉X
For curvature, need first fundamental form A. For horizontal vf’s
X, Y ,
Curvature downstairs is
A X Y = 1 2 V[X, Y ]
〈R ∗ (X ∗ , Y ∗ )Z ∗ , H〉 = 〈R(X, Y )Z, H〉
+ 2〈A X Y, A Z H〉 − 〈A Y Z, A X H〉 − 〈A Z X, A Y H〉
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 21