Polynomial Regression on Riemannian Manifolds

Have simple formula for cometric g ij (the kernel)
Parallel transport in terms of covectors, cometric:
d
dt β l = 1 2 g ilg in
,j g jm (α m β n − α n β m ) − 1 2 gmn ,l α m β n
Curvature more complicated (Mario’s Formula):
2R ursv = −g ur,sv − g rv,us + g rs,uv + g uv,rs + 2Γ rv
ρ Γ us
σ g ρσ − 2Γ rs
ρ Γ uv
σ g ρσ
+ g rλ,u g λµ g µv,s − g rλ,u g λµ g µs,v + g uλ,r g λµ g µs,v − g uλ,r g λµ g µv,s
+ g rλ,s g λµ g µv,u + g uλ,v g λµ g µs,r − g rλ,v g λµ g µs,u − g uλ,s g λµ g µv,r .
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 27