Embedded submanifolds: examples

Abstract

This chapter details how to work on several manifolds of practical interest, focusing on embedded submanifolds of linear spaces. It provides two tables which point to Manopt implementations of those manifolds, and to the various places in the book where it is explained how to work with products of manifolds. The manifolds detailed in this chapter include Euclidean spaces, unit spheres, the Stiefel manifold (orthonormal matrices), the orthogonal group and associated group of rotations, the manifold of matrices with a given size and rank and hyperbolic space in the hyperboloid model. It further discusses geometric tools for optimization on a manifold defined by (regular) constraints $h(x) = 0$ in general. That last section notably makes it possible to connect concepts from Riemannian optimization with classical concepts from constrained optimization in linear spaces, namely, Lagrange multipliers and KKT conditions under linear independence constraint qualifications (LICQ).