Directional derivatives on Kendall shape space

Abstract

We determine the expressions of the directional derivatives $\xi_{ij}\xi_{pq}$ regarding to the special tangential vector fields $\xi_{ij}$ on the shape space $\Sigma_{m}^{k}$ introduced by D.G. Kendall. The latter helps to associate a geometrical shape to arbitrary subsets of matrices in $\mathbb{R}^{m\times k}$ through the elimination of the effects of basic geometrical transformations. Then, the calculus that we need to perform here manipulates the inner products of the horizontal lifts
of the tangential vectors $\xi_{ij}$ since the shape space $\Sigma_{m}^{k}$ coincides with the quotient space of the unit hypersphere modulo specific special orthogonal group.