The Stiefel manifold, denoted as \( V_k(\mathbb{R}^n) \), is a mathematical object that describes the space of orthonormal k-frames in an n-dimensional Euclidean space \(\mathbb{R}^n\). More specifically, it consists of all matrices \( A \in \mathbb{R}^{n \times k} \) whose columns are orthonormal vectors in \(\mathbb{R}^n\).
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The real Stiefel manifold consists of ordered orthonormal -frames in . In particular, is naturally the unit tangent bundle of .