On Geodesics in Euclidean Shape Spaces

The overall geometry of the shape spaces ∑ m k , has been studied in [2,4], while the riemannian metric for each ∑ m k , is given in [5]. The aim of this paper is to explore the geodesic geometry of ∑ m k , especially when 2 ⩽ m ⩽ k−1. One of the main results is the identification of the cut-locus for each non-singular point in ∑ m k . We also obtain an extension of Casson's theorem concerning ∑ m m + 1 .