Stiefel Manifolds

Stiefel manifolds are geometric objects that represent all possible ways to choose a set of orthonormal vectors—meaning vectors that are perpendicular and of unit length—within a higher-dimensional space. For example, think of selecting a fixed number of perpendicular directions in three-dimensional space; the collection of all such sets forms a Stiefel manifold. These manifolds are important in mathematics and data science because they provide a structured way to analyze and optimize problems involving orientations, rotations, and subspace arrangements in multi-dimensional settings.